LeanAide GPT-4 documentation

Module: Mathlib.Analysis.Calculus.LHopital

HasDerivAt.lhopital_zero_nhds_right

theorem HasDerivAt.lhopital_zero_nhds_right : βˆ€ {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Ioi a), HasDerivAt f (f' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Ioi a), HasDerivAt g (g' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Ioi a), g' x β‰  0) β†’ Filter.Tendsto f (nhdsWithin a (Set.Ioi a)) (nhds 0) β†’ Filter.Tendsto g (nhdsWithin a (Set.Ioi a)) (nhds 0) β†’ Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Ioi a)) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioi a)) l

This theorem is a version of L'Hôpital's rule for approaching a real number from the right, specifically for functions with known derivatives. Given a real number `a`, a filter `l`, and four real-valued functions `f`, `f'`, `g`, `g'`, the theorem states the following conditions: if for all `x` in the neighborhood within the set of real numbers greater than `a` (i.e., the interval (a, +∞)), `f` has the derivative `f'` at `x`, `g` has the derivative `g'` at `x`, and `g'(x)` is not equal to zero, and if the functions `f` and `g` tend to zero as `x` tends to `a` from the right, and the quotient of `f'` and `g'` tends to `l` as `x` tends to `a` from the right, then the quotient of `f` and `g` also tends to `l` as `x` tends to `a` from the right. In mathematical terms, if $\lim_{{x \to a^+}} f(x) = \lim_{{x \to a^+}} g(x) = 0$, $\lim_{{x \to a^+}} \frac{{f'(x)}}{{g'(x)}} = l$, and $g'(x) \neq 0$ for $x$ in a neighborhood of $a$ on the right, then $\lim_{{x \to a^+}} \frac{{f(x)}}{{g(x)}} = l$.

More concisely: If functions `f` and `g` have derivatives `f'` and `g'` respectively, `g'(x) β‰  0` for `x > a`, and as `x -> a+`, `f(x) / g(x) -> l = (f'(x) / g'(x)`) where `lim (x -> a+) f(x) = lim (x -> a+) g(x) = 0`.

HasDerivAt.lhopital_zero_nhds

theorem HasDerivAt.lhopital_zero_nhds : βˆ€ {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhds a, HasDerivAt f (f' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhds a, HasDerivAt g (g' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhds a, g' x β‰  0) β†’ Filter.Tendsto f (nhds a) (nhds 0) β†’ Filter.Tendsto g (nhds a) (nhds 0) β†’ Filter.Tendsto (fun x => f' x / g' x) (nhds a) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a {a}ᢜ) l

This is the statement of **L'HΓ΄pital's rule** for limits of real functions, in the context where the derivative of the functions is explicitly considered. More precisely, if `f` and `g` are two functions which are differentiable in a neighborhood of a point `a`, and their derivatives `f'` and `g'` satisfy that `g'` is not zero in this neighborhood, and both `f` and `g` tend to `0` when `x` tends to `a`, then if the ratio of `f'` and `g'` tends to a limit `l` when `x` tends to `a`, it follows that the ratio of `f` and `g` also tends to `l` when `x` tends to `a`, but excluding the point `a` itself from the domain of consideration.

More concisely: If functions `f` and `g` are differentiable near `a` with non-zero derivatives, and `f/g` has a limit as `x` approaches `a`, then `f` and `g` have the same limit as `x` approaches `a`. (Excluding `a` from the domain of `x`.)

deriv.lhopital_zero_nhds_left

theorem deriv.lhopital_zero_nhds_left : βˆ€ {a : ℝ} {l : Filter ℝ} {f g : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Iio a), DifferentiableAt ℝ f x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Iio a), deriv g x β‰  0) β†’ Filter.Tendsto f (nhdsWithin a (Set.Iio a)) (nhds 0) β†’ Filter.Tendsto g (nhdsWithin a (Set.Iio a)) (nhds 0) β†’ Filter.Tendsto (fun x => deriv f x / deriv g x) (nhdsWithin a (Set.Iio a)) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Iio a)) l

This theorem is a version of **L'HΓ΄pital's rule** for approaching a real number from the left, using derivatives. It states that for any real number `a`, any filter `l`, and any two real-valued functions `f` and `g`, if `f` is differentiable at every point in the left-infinite right-open interval ending at `a` and the derivative of `g` is never zero in that interval, and both `f` and `g` tend towards zero as they approach `a` from the left, and the ratio of the derivatives of `f` and `g` in that interval tends towards `l`, then the ratio of `f` and `g` themselves also tends towards `l` as they approach `a` from the left. In simpler terms, if `f` and `g` are going to zero and `g` is not flat, then the limit of the ratio `f/g` is the same as the limit of the ratio of their derivatives.

More concisely: If functions `f` and `g` are differentiable with non-zero derivatives in the left-infinite right-open interval approaching `a`, `f/g` tends towards the limit of `(f'/g')` as `x` approaches `a` from the left.

deriv.lhopital_zero_atTop

theorem deriv.lhopital_zero_atTop : βˆ€ {l : Filter ℝ} {f g : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in Filter.atTop, DifferentiableAt ℝ f x) β†’ (βˆ€αΆ  (x : ℝ) in Filter.atTop, deriv g x β‰  0) β†’ Filter.Tendsto f Filter.atTop (nhds 0) β†’ Filter.Tendsto g Filter.atTop (nhds 0) β†’ Filter.Tendsto (fun x => deriv f x / deriv g x) Filter.atTop l β†’ Filter.Tendsto (fun x => f x / g x) Filter.atTop l

This is a statement of **L'HΓ΄pital's rule** in the context of limits as the independent variable approaches positive infinity. The theorem asserts that given two real-valued functions `f` and `g` such that both `f` and `g` approach zero as the independent variable goes to positive infinity, and for all sufficiently large real values `x`, `f` is differentiable and the derivative of `g` is non-zero. If the limit of the ratio of the derivatives of `f` and `g` as `x` goes to positive infinity exists and equals `l`, then the limit of the ratio of `f(x)` and `g(x)` as `x` goes to positive infinity also exists and equals `l`. This theorem is a powerful tool for calculating limits involving ratios of functions where both the numerator and denominator approach zero as the variable goes to infinity.

More concisely: If functions `f` and `g` approach 0 as `x` goes to positive infinity, `f` is differentiable with non-zero derivative for large `x`, and the limit of `(f'/g')` as `x` goes to infinity exists, then the limit of `(f/g)` as `x` goes to infinity exists and equals that limit.

deriv.lhopital_zero_atBot

theorem deriv.lhopital_zero_atBot : βˆ€ {l : Filter ℝ} {f g : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in Filter.atBot, DifferentiableAt ℝ f x) β†’ (βˆ€αΆ  (x : ℝ) in Filter.atBot, deriv g x β‰  0) β†’ Filter.Tendsto f Filter.atBot (nhds 0) β†’ Filter.Tendsto g Filter.atBot (nhds 0) β†’ Filter.Tendsto (fun x => deriv f x / deriv g x) Filter.atBot l β†’ Filter.Tendsto (fun x => f x / g x) Filter.atBot l

This theorem is a version of L'HΓ΄pital's rule for the case where the limit approaches negative infinity. It states that if for all sufficiently negative real numbers `x`, the functions `f` and `g` are differentiable at `x`, and the derivative of `g` at `x` is not zero, and furthermore if the functions `f` and `g` both tend to zero as `x` tends to negative infinity, then if the limit as `x` tends to negative infinity of the ratio of the derivatives of `f` and `g` at `x` exists and equals some real number `l`, then the limit as `x` tends to negative infinity of the ratio of `f(x)` to `g(x)` also exists and equals `l`. In mathematical terms, this is usually written as if $\lim_{x \to -\infty} f'(x) / g'(x) = l$, then $\lim_{x \to -\infty} f(x) / g(x) = l$, assuming that $f$ and $g$ are differentiable, $g'(x) \neq 0$, and $\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} g(x) = 0$.

More concisely: If functions `f` and `g` are differentiable for all sufficiently large negative real numbers `x`, `g'(x)` never equals zero, `f(x)` and `g(x)` both tend to zero as `x` goes to negative infinity, and the limit of `f'(x) / g'(x)` as `x` goes to negative infinity exists, then the limit of `f(x) / g(x)` as `x` goes to negative infinity also exists and equals the limit of `f'(x) / g'(x)`.

HasDerivAt.lhopital_zero_nhds'

theorem HasDerivAt.lhopital_zero_nhds' : βˆ€ {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhdsWithin a {a}ᢜ, HasDerivAt f (f' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a {a}ᢜ, HasDerivAt g (g' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a {a}ᢜ, g' x β‰  0) β†’ Filter.Tendsto f (nhdsWithin a {a}ᢜ) (nhds 0) β†’ Filter.Tendsto g (nhdsWithin a {a}ᢜ) (nhds 0) β†’ Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a {a}ᢜ) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a {a}ᢜ) l

This theorem is a version of L'HΓ΄pital's rule for real numbers, which deals with the case where a function approaches a given value, using the `HasDerivAt` formulation. It states that for any real number `a` and any filter `l`, and given two functions `f` and `g` along with their derivatives `f'` and `g'`, if the following conditions are satisfied: - In the neighborhood of `a` excluding `a` itself, `f` and `g` have derivatives `f'` and `g'`, respectively. - Also in the neighborhood of `a` excluding `a` itself, `g'` is non-zero. - The functions `f` and `g` both tend to `0` as `x` approaches `a` (excluding `a` itself). - The function `f'/g'` tends to `l` as `x` approaches `a` (excluding `a` itself). Then, it can be concluded that the function `f/g` also tends to `l` as `x` approaches `a` (excluding `a` itself). This theorem does not require any conditions at the point `a` itself.

More concisely: If real functions `f` and `g` have derivatives `f'` and `g'` in a neighborhood of `a` excluding `a` itself, `g'` is never zero in that neighborhood, `f`, `g` approach 0 as `x` approaches `a`, and `f'/g'` approaches a filter `l` as `x` approaches `a`, then `f/g` approaches `l` as `x` approaches `a`.

deriv.lhopital_zero_nhds

theorem deriv.lhopital_zero_nhds : βˆ€ {a : ℝ} {l : Filter ℝ} {f g : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhds a, DifferentiableAt ℝ f x) β†’ (βˆ€αΆ  (x : ℝ) in nhds a, deriv g x β‰  0) β†’ Filter.Tendsto f (nhds a) (nhds 0) β†’ Filter.Tendsto g (nhds a) (nhds 0) β†’ Filter.Tendsto (fun x => deriv f x / deriv g x) (nhds a) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a {a}ᢜ) l

**L'HΓ΄pital's rule** in the context of derivative (`deriv`) is as follows: Given a real number `a` and a filter `l` on real numbers, and two real-valued functions `f` and `g`, if `f` is differentiable at every point in a neighborhood of `a`, and the derivative of `g` at every point in that neighborhood is not zero, and if `f` and `g` both tend to zero as they approach `a`, and the ratio of the derivatives of `f` and `g` tends to `l` as we approach `a`, then the ratio `f(x) / g(x)` also tends to `l` as `x` approaches `a` from either side, excluding `a` itself.

More concisely: If `f` and `g` are real-valued functions, `f` is differentiable at every point near `a`, `g` is never zero near `a`, `f(x) / g(x)` tends to `l` as `x` approaches `a` if `f'(x) / g'(x)` tends to `l` as `x` approaches `a`, and both `f` and `g` tend to 0 as `x` approaches `a`, then `f(x) / g(x)` tends to `l` as `x` approaches `a` (excluding `a`).

HasDerivAt.lhopital_zero_nhds_left

theorem HasDerivAt.lhopital_zero_nhds_left : βˆ€ {a : ℝ} {l : Filter ℝ} {f f' g g' : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Iio a), HasDerivAt f (f' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Iio a), HasDerivAt g (g' x) x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Iio a), g' x β‰  0) β†’ Filter.Tendsto f (nhdsWithin a (Set.Iio a)) (nhds 0) β†’ Filter.Tendsto g (nhdsWithin a (Set.Iio a)) (nhds 0) β†’ Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Iio a)) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Iio a)) l

This theorem is a version of L'HΓ΄pital's rule for a point approached from the left. It states that if for every point x in a left-sided neighborhood of a real number a, the functions f and g have derivatives f' and g' respectively, g' is non-zero, f and g tend to 0, and the quotient of f' to g' tends to some limit, then the quotient of f to g also tends to the same limit. In mathematical terms, as x approaches a from the left, if f'(x)/g'(x) tends to a limit, then f(x)/g(x) tends to the same limit provided the derivatives exist and f(x) and g(x) tend to 0.

More concisely: If limits exist and are finite, then lim (f(x)/g(x)) = lim (f'(x)/g'(x)) as x approaches a from the left, provided f', g' are continuous at a and g'(a) β‰  0.

HasDerivAt.lhopital_zero_right_on_Ioo

theorem HasDerivAt.lhopital_zero_right_on_Ioo : βˆ€ {a b : ℝ}, a < b β†’ βˆ€ {l : Filter ℝ} {f f' g g' : ℝ β†’ ℝ}, (βˆ€ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) β†’ (βˆ€ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x) β†’ (βˆ€ x ∈ Set.Ioo a b, g' x β‰  0) β†’ Filter.Tendsto f (nhdsWithin a (Set.Ioi a)) (nhds 0) β†’ Filter.Tendsto g (nhdsWithin a (Set.Ioi a)) (nhds 0) β†’ Filter.Tendsto (fun x => f' x / g' x) (nhdsWithin a (Set.Ioi a)) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioi a)) l

This is a statement of L'Hopital's rule for the case where the point of interest is the left endpoint of an open interval. Specifically, the theorem states that for any two real-valued functions `f` and `g` that are differentiable on an open interval `(a, b)`, if `f` and `g` both approach `0` as `x` approaches `a` from the right (in other words, within the interval `(a, ∞)`), and the derivative of `g` is never `0` in `(a, b)`, then the limits as `x` approaches `a` from the right of the ratio of the derivatives of `f` and `g` and the ratio of `f` and `g` are the same. Note that L'Hopital's rule is a method for evaluating limits of ratios of two functions that approach `0` or infinity.

More concisely: If functions `f` and `g` are differentiable on an open interval `(a, b)`, `f(x)` and `g(x)` approach 0 as `x` approaches `a` from the right, and `g'(x)` never equals 0 in `(a, b)`, then `(f'/g')(a) = (f/g)(a)`.

deriv.lhopital_zero_nhds'

theorem deriv.lhopital_zero_nhds' : βˆ€ {a : ℝ} {l : Filter ℝ} {f g : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhdsWithin a {a}ᢜ, DifferentiableAt ℝ f x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a {a}ᢜ, deriv g x β‰  0) β†’ Filter.Tendsto f (nhdsWithin a {a}ᢜ) (nhds 0) β†’ Filter.Tendsto g (nhdsWithin a {a}ᢜ) (nhds 0) β†’ Filter.Tendsto (fun x => deriv f x / deriv g x) (nhdsWithin a {a}ᢜ) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a {a}ᢜ) l

This theorem is a version of **L'HΓ΄pital's rule**, a principle used to evaluate limits of functions, applied for a real number and using the derivative form. It does not make any assumptions about the situation at point `a`. More specifically, it states that given a real number `a`, a filter `l` on the reals, and two real-valued functions `f` and `g`, if the following conditions are met: 1. The function `f` is differentiable at every point in a neighborhood within the set of all real numbers except `a`, 2. The derivative of `g` at every point in the same neighborhood is non-zero, 3. The limit of `f` as `x` approaches `a` (without `x` being equal to `a`) is 0, 4. Similarly, the limit of `g` as `x` approaches `a` (without `x` being equal to `a`) is also 0, 5. The limit of the ratio of the derivatives of `f` and `g` as `x` approaches `a` (without `x` being equal to `a`) is equal to the filter `l`. Then, the limit of the ratio of `f(x)` and `g(x)` as `x` approaches `a` (without `x` being equal to `a`) is also equal to the filter `l`.

More concisely: If functions `f` and `g` are differentiable around `a` except at `a` with non-zero derivatives, `lim (f(x)/g(x)) = l <- 0` if and only if `lim (f'(x)/g'(x)) = l`.

deriv.lhopital_zero_nhds_right

theorem deriv.lhopital_zero_nhds_right : βˆ€ {a : ℝ} {l : Filter ℝ} {f g : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Ioi a), DifferentiableAt ℝ f x) β†’ (βˆ€αΆ  (x : ℝ) in nhdsWithin a (Set.Ioi a), deriv g x β‰  0) β†’ Filter.Tendsto f (nhdsWithin a (Set.Ioi a)) (nhds 0) β†’ Filter.Tendsto g (nhdsWithin a (Set.Ioi a)) (nhds 0) β†’ Filter.Tendsto (fun x => deriv f x / deriv g x) (nhdsWithin a (Set.Ioi a)) l β†’ Filter.Tendsto (fun x => f x / g x) (nhdsWithin a (Set.Ioi a)) l

This theorem is a version of **L'HΓ΄pital's Rule** for differentiable functions approaching a real number from the right. It states that for two real-valued functions `f` and `g`, if `f` is differentiable and `g`'s derivative is nonzero in a neighborhood within a right-open interval around a real number `a`, and if both `f` and `g` tend towards 0 in the same neighborhood, then the limit as `x` approaches `a` from the right of the ratio of `f(x)` to `g(x)` is equal to the limit of the ratio of their derivatives in the same neighborhood, given that the latter limit exists.

More concisely: If `f` and `g` are real-valued functions that are differentiable, with nonzero derivative for `g` in a right neighborhood of `a`, and both functions approach 0 in that neighborhood, then `lim (f(x)/g(x)) = lim (f'(x)/g'(x))` as `x -> a` from the right, where the derivatives exist.

HasDerivAt.lhopital_zero_atTop

theorem HasDerivAt.lhopital_zero_atTop : βˆ€ {l : Filter ℝ} {f f' g g' : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in Filter.atTop, HasDerivAt f (f' x) x) β†’ (βˆ€αΆ  (x : ℝ) in Filter.atTop, HasDerivAt g (g' x) x) β†’ (βˆ€αΆ  (x : ℝ) in Filter.atTop, g' x β‰  0) β†’ Filter.Tendsto f Filter.atTop (nhds 0) β†’ Filter.Tendsto g Filter.atTop (nhds 0) β†’ Filter.Tendsto (fun x => f' x / g' x) Filter.atTop l β†’ Filter.Tendsto (fun x => f x / g x) Filter.atTop l

This is an implementation of L'HΓ΄pital's rule for functions approaching infinity. The theorem states that if a function `f` and another function `g`, along with their derivatives `f'` and `g'`, both have limits of zero as `x` approaches infinity (denoted by `Filter.atTop`), and the derivative of `g` is never zero, then if the limit of the ratio of the derivatives `f' / g'` as `x` approaches infinity exists and equals `l`, the limit of the ratio of the functions `f / g` as `x` approaches infinity will also be `l`. In more formal terms, if: $\forall x \rightarrow \infty$, $f'(x)$ is the derivative of $f(x)$, $g'(x)$ is the derivative of $g(x)$, $g'(x) \neq 0$, $\lim_{x\rightarrow\infty} f(x) = 0$, $\lim_{x\rightarrow\infty} g(x) = 0$, and $\lim_{x\rightarrow\infty} \frac{f'(x)}{g'(x)} = l$, then $\lim_{x\rightarrow\infty} \frac{f(x)}{g(x)} = l$.

More concisely: If functions `f` and `g` have derivatives `f'` and `g'` with `g' never equal to zero, and if `f`, `g`, `f'/g'` all approach zero as `x` approaches infinity, then `f/g` approaches the same limit as `f'/g'`.

HasDerivAt.lhopital_zero_atBot

theorem HasDerivAt.lhopital_zero_atBot : βˆ€ {l : Filter ℝ} {f f' g g' : ℝ β†’ ℝ}, (βˆ€αΆ  (x : ℝ) in Filter.atBot, HasDerivAt f (f' x) x) β†’ (βˆ€αΆ  (x : ℝ) in Filter.atBot, HasDerivAt g (g' x) x) β†’ (βˆ€αΆ  (x : ℝ) in Filter.atBot, g' x β‰  0) β†’ Filter.Tendsto f Filter.atBot (nhds 0) β†’ Filter.Tendsto g Filter.atBot (nhds 0) β†’ Filter.Tendsto (fun x => f' x / g' x) Filter.atBot l β†’ Filter.Tendsto (fun x => f x / g x) Filter.atBot l

This theorem is a version of L'Hôpital's rule for when a real-valued function approaches negative infinity (`-∞`). Given two real-valued functions `f` and `g` and their derivatives `f'` and `g'` respectively, if `f` and `g` both approach `0` as `x` approaches `-∞`, and `g'` is not `0` near `-∞`, and the ratio of `f'` and `g'` approaches a real filter `l` as `x` approaches `-∞`, then the ratio of `f` and `g` also approaches the same filter `l` as `x` approaches `-∞`. This theorem essentially states that under these conditions, the limit of the ratio of `f` and `g` as `x` approaches `-∞` exists and is the same as the limit of the ratio of their derivatives.

More concisely: If real-valued functions `f` and `g` approach 0 as `x` approaches -∞, `g'` never equals 0 near -∞, and the limit as `x -> -∞` of `f'/g'` exists, then the limit as `x -> -∞` of `f/g` equals that limit.