3.8: L'Hôpital's Rule

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    3.8 L'Hôpital's Rule

    Learning Objectives
    • Identify indeterminate forms such as 0/0 or ∞/∞ in limits, and apply L'Hôpital's rule to evaluate these limits by differentiating the numerator and denominator until a determinate form is obtained.

    Consider the following limit:

    \[
    \lim_{x \to -3} \dfrac{x^2 + 4x + 3}{x^2 - 9}
    \]

    (a) How do we evaluate the limit above?

    (b) Once you identify the way to evaluate it, then evaluate the limit.

    (c) Let \( f(x) = x^2 + 4x + 3 \) and \( g(x) = x^2 - 9 \). Find the linearization of \( f(x) \) at \( a = -3 \), denoted \( L_f(x) \), and the linearization of \( g(x) \) at \( a = -3 \), denoted \( L_g(x) \).

    (d) Evaluate the limit: \( \lim_{x \to -3} \dfrac{L_f(x)}{L_g(x)} \)

    (e) Compare the results from parts (b) and (d).

    Now, let's suppose that \( f \) and \( g \) are differentiable functions at \( x = a \) such that:

    \[
    \lim_{x \to a} f(x) = 0 = \lim_{x \to a} g(x),
    \]

    and both \( f'(x) \) and \( g'(x) \) are continuous at \( a \), with \( g'(a) \neq 0 \). If \( x \) is near \( a \), we can use linear approximation:

    \[
    f(x) \approx f(a) + f'(a)(x - a)
    \quad \text{and} \quad
    g(x) \approx g(a) + g'(a)(x - a)
    \]

    Therefore, the quotient can be approximated as:

    \[
    \frac{f(x)}{g(x)} \approx \frac{f(a) + f'(a)(x - a)}{g(a) + g'(a)(x - a)}
    \]

    Two functions y = f(x) and y = g(x) are drawn such that they cross at a point above x = a. The linear approximations of these two functions y = f(a) + f’(a)(x – a) and y = g(a) + g’(a)(x – a) are also drawn.

    Figure \(\PageIndex{13}\): The graph of two functions and their linear approximations. From OpenStax Calculus Volume 1, licensed under CC-BY-NC-SA.

    Both \( f \) and \( g \) are continuous at \( a \) because they are differentiable at \( a \), and differentiability implies continuity.

    Therefore:

    \[
    f(a) = \lim_{x \to a} f(x) = 0 \quad \text{and} \quad g(a) = \lim_{x \to a} g(x) = 0
    \]

    If \( g'(a) \neq 0 \), then we can apply linear approximations:

    \[
    \lim_{x \to a} \frac{f(x)}{g(x)} =
    \lim_{x \to a} \frac{f(a) + f'(a)(x - a)}{g(a) + g'(a)(x - a)} =
    \lim_{x \to a} \frac{f'(a)(x - a)}{g'(a)(x - a)} =
    \lim_{x \to a} \frac{f'(a)}{g'(a)}
    \]

    So,

    \[
    \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)}
    \]

    Note

    Observe that we are assuming that \(f'\) and \(g'\) are continuous.

    Theorem \(\PageIndex{7}\) L'Hôpital Rule- Indeterminate Form \(\dfrac{0}{0}\).

    Suppose that \( f \) and \( g \) are differentiable on an open interval containing \( a \), except possibly at \( a \), and that:

    \[
    \lim_{x \to a} f(x) = 0 \quad \text{and} \quad \lim_{x \to a} g(x) = 0,
    \]

    with \( g'(x) \neq 0 \) near \( a \). Then:

    \[
    \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)},
    \]

    provided the limit on the right exists.

    The same result applies for one-sided limits or limits as \( x \to \pm \infty \).

    Example \(\PageIndex{26}\)

    Evaluate the following limits:

    1. \( \displaystyle \lim_{x \to 1} \frac{x^3 - 4x^2 + 3}{x^2 - 1} \)
    2. \( \displaystyle \lim_{x \to 0} \frac{\sin x}{x} \)
    3. \( \displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} \)
    4. \( \displaystyle \lim_{x \to 0} \frac{e^x - 1}{\sin(\pi x)} \)
    5. \( \displaystyle \lim_{x \to 0} \frac{\tan x}{4x} \)
    6. \( \displaystyle \lim_{x \to \infty} \frac{e^{5/x} - 1}{1/x^2} \)
    Theorem \(\PageIndex{8}\) L'Hôpital Rule - Indeterminate Form \(\dfrac{\infty}{\infty}\)

    Suppose that \( f \) and \( g \) are differentiable on an open interval containing \( a \), except possibly at \( a \). If

    \[
    \lim_{x \to a} f(x) = \infty \quad \text{or } -\infty, \quad \text{and} \quad \lim_{x \to a} g(x) = \infty \quad \text{or } -\infty,
    \]

    then

    \[
    \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)},
    \]

    provided the limit on the right exists.

    This result also holds for one-sided limits and limits as \( x \to \pm \infty \).

    Example \(\PageIndex{27}\)

    Evaluate the following limits:

    1. \( \displaystyle \lim_{x \to \infty} \frac{x^2 + 8x}{3x + 1} \)
    2. \( \displaystyle \lim_{x \to \infty} x e^{-x^2} \)
    3. \( \displaystyle \lim_{x \to \infty} \frac{\ln x}{2\sqrt{x}} \)

    Other Indeterminate Forms

    Example \(\PageIndex{28}\)

    Evaluate the following limits.

    1. \( \displaystyle \lim_{x \to 0^+} x \ln x \). Indeterminate form: \( 0 \cdot \infty \)
    2. \( \displaystyle \lim_{x \to \pi/2^+} (\sec x - \tan x) \). Indeterminate form: \( \infty - \infty \)
    3. \( \displaystyle \lim_{x \to \infty} x^{1/\sqrt{x}} \). Indeterminate form: \( \infty^0 \)
    4. \( \displaystyle \lim_{x \to 0^+} x^{3/x} \). Indeterminate form: \( 0^\infty \)
    5. \( \displaystyle \lim_{x \to 0^+} (1 + \sin 6x)^{1/x} \). Indeterminate form: \( 1^\infty \)
    6. \( \displaystyle \lim_{x \to 0^+} x^x \). Indeterminate form: \( 0^0 \)

    This page titled 3.8: L'Hôpital's Rule was last modified on Sun, 07 Dec 2025 01:10:58 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Kevin Palencia.