7.7E: Exercises for L'Hôpital's Rule (2024)

  1. Last updated
  2. Save as PDF
  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vectorC}[1]{\textbf{#1}}\)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    Terms and Concepts

    1. List the different indeterminate forms described in this section.

    The forms \(0⋅∞, ∞−∞, 1^∞, ∞^0\), and \(0^0\) are all considered indeterminate.

    2. List similar looking forms that are not indeterminate.

    Among others, the forms \(∞⋅∞, ∞+∞, -∞−∞, 0^∞, 1^0\), and \(∞^∞\) are not considered indeterminate, as these limits can be determined clearly.

    3. T/F: l'Hôpital's Rule states that \(\frac{d}{dx} \left ( \frac{f(x)}{g(x)}\right ) = \frac{f'(x)}{g'(x)}\).

    False. L'Hôpital's Rule is a method for taking limits of rational functions in certain cases. It does not replace the Quotient Rule when taking the derivative of these rational functions.

    4. Explain what the indeterminate form "\(1^{\infty}\)" means. Why is it indeterminate?

    When a limit has the form "\(1^{\infty}\)", this means that the function in the base of the exponent is approaching \(1\), while the function in the exponent is approaching \(\infty\). It's indeterminate, since if the base function is approaching 1, but always is less than 1, then the limit could be 0, while if the base function were approaching 1, but always is greater than 1, the limit could be \(\infty\). But since this uncertainty exists, the limit could, in fact, be anything.

    5. Explain why limits of the form \(\infty - \infty\) are indeterminate.

    Limits with this form depend on the relative speed with which the two terms are approaching \(\infty\). If the first term approaches \(\infty\) faster than the second term, the limit would be \(\infty\). If the second term approaches \(\infty\) faster than the first term, the limit would be \(-\infty\). But if they both appraoch \(\infty\) at about the same rates, the limit could be anything!

    6. Fill in the blanks" The Quotient Rule is applied to \(\frac{f(x)}{g(x)}\) when taking its _____; l'Hôpital's Rule is applied when taking _______ of \(\frac{f(x)}{g(x)}\) when the form is _______ or _______.

    derivative; limits; \(\dfrac{0}{0}\) or \(\dfrac{\pm\infty}{\pm\infty}\)

    7. Create (but do not evaluate) a limit that initially has the form "\(\infty^0\)".

    8. Create a function \(f(x)\) such that \(\lim\limits_{x\to1}f(x)\) initially has the form "\(0^0\)".


    For exercises 1 - 6, evaluate the limit.

    1) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{e^x}{x}\).

    2) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k}\).

    \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k} \quad = \quad ∞\)

    3) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{\ln x}{x^k}\).

    4) Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}\).

    \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}\)

    5. Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^3−a^3}\).

    6. Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n}\).

    \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}\)

    For exercises 7 - 11, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.

    7) \(\displaystyle \lim_{x→0^+}x^2\ln x\)

    8) \(\displaystyle \lim_{x→∞}x^{1/x}\)

    Cannot apply directly; use logarithms

    9) \(\displaystyle \lim_{x→0}x^{2/x}\)

    10) \(\displaystyle \lim_{x→0}\frac{x^2}{1/x}\)

    Cannot apply directly; rewrite as \(\displaystyle \lim_{x→0}x^3\)

    11) \(\displaystyle \lim_{x→∞}\frac{e^x}{x}\)

    For exercises 12 - 44, evaluate the limits with either L’Hôpital’s rule or previously learned methods.

    12) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}\)

    \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3} \quad = \quad 6\)

    13) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}\)

    14) \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}\)

    \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2\)

    15) \(\displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}\)

    16) \(\displaystyle \lim_{x→π}\frac{x−π}{\sin x}\)

    \(\displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1\)

    17) \(\displaystyle \lim_{x→1}\frac{x−1}{\sin x}\)

    18) \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}\)

    \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n\)

    19) \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}\)

    20) \(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}\)

    \(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}\)

    21) \(\displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}\)

    22) \(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}\)

    \(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}\)

    23) \(\displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}\)

    24) \(\displaystyle \lim_{x→1}\frac{x-1}{\ln x}\)

    \(\displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1\)

    25) \(\displaystyle \lim_{x→0}\,(x+1)^{1/x}\)

    26) \(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}\)

    \(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}\)

    27) \(\displaystyle \lim_{x→0^+}x^{2x}\)

    28) \(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)\)

    \(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1\)

    29) \(\displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}\)

    30) \(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right)\)

    \(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0\)

    31) \(\displaystyle \lim_{x→∞}(x−e^x)\)

    32) \(\displaystyle \lim_{x→∞}x^2e^{−x}\)

    \(\displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0\)

    33) \(\displaystyle \lim_{x\to 1^+} \left[\frac{1}{\ln x}-\frac{1}{1-x}\right]\)

    34) \(\displaystyle \lim_{x\to 3^+} \left[\frac{5}{x^2-9}-\frac{x}{x-3}\right]\)

    \(\displaystyle \lim_{x\to 3^+} \left[\frac{5}{x^2-9}-\frac{x}{x-3}\right] = \lim_{x\to 3^+} \frac{5-x^2-3x}{x^2-9} \quad = \quad -∞\)

    35) \(\displaystyle \lim_{x\to \infty} \frac{\sqrt{2x^2-3}}{x+2}\)

    L’Hôpital’s rule fails to help us find this limit, although the form seems appropriate. But you can evaluate this limit using techniques you learned earlier in calculus.

    36) \(\displaystyle \lim_{x\to \infty} \left(\frac{x+7}{x+3}\right)^{x}\)

    \(\displaystyle \lim_{x\to \infty} \left(\frac{x+7}{x+3}\right)^{x} \quad = \quad e^{4}\)

    37) \(\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}\)

    38) \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}\)

    \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1\)

    39) \(\displaystyle \lim_{x→π/4}(1−\tan x)\cot x\)

    40) \(\displaystyle \lim_{x→∞}xe^{1/x}\)

    \(\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞\)

    41) \(\displaystyle \lim_{x→0}x^{1/\cos x}\)

    42) \(\displaystyle \lim_{x→0^+}x^{1/x}\)

    \(\displaystyle \lim_{x→0^+}x^{1/x} \quad = \quad 0\)

    43) \(\displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x\)

    44) \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x\)

    \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}\)

    For exercises 45 - 54, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.

    45) [T] \(\displaystyle \lim_{x→0}\frac{e^x−1}{x}\)

    46) [T] \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)\)

    \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0\)

    47) [T] \(\displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}\)

    48) [T] \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}\)

    \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1\)

    49) [T] \(\displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}\)

    50) [T] \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}\)

    \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0\)

    51) [T] \(\displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)\)

    52) [T] \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right)\)

    \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1\)

    53) [T] \(\displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}\)

    54) [T] \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}\)

    \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2\)


    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensedwith a CC-BY-SA-NC4.0license. Download for free at http://cnx.org.

    • Gregory Hartman (Virginia Military Institute).Contributions were made by Troy Siemers andDimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content iscopyrighted by a Creative CommonsAttribution - Noncommercial (BY-NC) License.http://www.apexcalculus.com/

    • Terms and Concepts1-4, and 6-8 were adapted from Apex Calculus by Paul Seeburger (Monroe Community College)
    • Terms and Concepts Problem 5 was created by Paul Seeburger
    • Problems 33 and 34 from Apex Calculus. Solution to 34 by Paul Seeburger
    • Problems 35and 36 and solutions for Terms and Concepts Problems 1-6 and for Problem 36 were added by Paul Seeburger
    7.7E: Exercises for L'Hôpital's Rule (2024)


    What are the numerical methods for ordinary differential equations? ›

    Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.

    How to solve partial differential equations numerically? ›

    Overview of methods
    1. Finite difference method.
    2. Method of lines.
    3. Finite element method.
    4. Gradient discretization method.
    5. Finite volume method.
    6. Spectral method.
    7. Meshfree methods.
    8. Domain decomposition methods.

    What is Euler's method calculus? ›

    Euler's method is a numerical tool for approximating values for solutions of differential equations.

    What is a numerical solution? ›

    In mathematics, some problems can be solved analytically and numerically. An analytical solution involves framing the problem in a well-understood form and calculating the exact solution. A numerical solution means making guesses at the solution and testing whether the problem is solved well enough to stop.

    What is the easiest numerical method? ›

    Euler's Method is one of the simplest and oldest numerical methods for approximating solutions to differential equations that cannot be solved with a nice formula.

    What is the best method to solve a differential equation? ›

    Ans: If the differential equation is of the form f ( x ) d x = g ( y ) d y , where and are functions of and only. Then we say that the variables are separable in the differential equation. Thus, integrating both sides of the equation, we get its general solution.

    How do you solve numerical differentiation? ›

    8.1 Numerical Differentiation

    To compute dy/dx, we first replace the exact relation y = f(x) by the best interpolating polynomial y = (x) and then differentiate the latter as many times as we desire. The choice of the interpola- tion formula to be used, will depend on the assigned value of x at which dy/dx is desired.

    What are the numerical methods for solving fractional differential equations? ›

    In this article, two numerical techniques namely, the hom*otopy perturbation method and the matrix approach method have been proposed and implemented to solve fractional differential equations. The accuracy and the validity of these techniques are tested with some numerical examples.

    Did NASA use Euler's method? ›

    Summary --- Katherine Johnson (NASA 1969)

    As told in the book (and movie) Hidden Figures, Katherine Johnson led the team of African-American women who did the actual calculation of the necessary trajectory from the earth to the moon for the US Apollo space program. They used Euler's method to do this.

    Is Euler's method old or new math? ›

    First off, Euler's Method is indeed pretty old, if not exactly ancient. It was developed by Leonhard Euler (pronounced oy-ler), a prolific Swiss mathematician who lived 1707-1783.

    How to solve Euler equation? ›

    The basic approach to solving Euler equations is similar to the approach used to solve constant-coefficient equations: assume a particular form for the solution with one constant “to be determined”, plug that form into the differential equation, simplify and solve the resulting equation for the constant, and then ...

    How do you solve Numericals? ›

    Simplify a complex numerical by chunking it and making diagrams. This will ease the process of selecting the correct equation and ending up with a correct answer. Memorize all relevant equations and the conditions in which they are best applicable.

    Are numerical methods hard? ›

    Numerical Analysis is likely to be one of the most intensive problem-solving courses an undergraduate student will take.

    What are the steps for solving a numerical problem? ›

    12.4: The 5-Step Method of Solving Applied Problems
    1. Let x (or some other letter) represent the unknown quantity.
    2. Translate the English to mathematics and form an equation.
    3. Solve this equation.
    4. Check this result by substituting it into the original statement of the problem.
    5. Write a conclusion.
    Jun 11, 2021

    What are the 4 types of ordinary differential equations? ›

    The types of DEs are partial differential equation, linear and non-linear differential equations, hom*ogeneous and non-hom*ogeneous differential equation.

    What is the numerical method of differentiation? ›

    There are three methods of computing Numerical Differentiation: Forward, Backward and Central difference. While the Forward and Backward methods are easy and involve simple mathematics respectively, they aren't very accurate. The Central Difference method is the most accurate but involves complex maths.

    What are the numerical methods for solving equations? ›

    Other commonly used numerical methods for solving equations include Newton's method, the bisection method, and the secant method. These methods use iterative approaches to finding the solution where each iteration yields a better approximation than the previous.

    What are the one step methods for ordinary differential equations? ›

    3.3. 1 One-step methods
    1styj+1 = yj + hf(yj,tj)
    2ndy j + 1 = y j + h f ( y j + h 2 f ( y j , t j ) , t k + h 2 )
    3rdy j + 1 = y j + h 4 ( k 1 + 3 k 3 ) k 1 = f ( y j , t j ) k 2 = f ( y j + h k 1 3 , t j + h 3 ) k 3 = f ( y j + 2 h k 2 3 , t j + 2 h 3 )
    1 more row

    Top Articles
    Latest Posts
    Article information

    Author: Van Hayes

    Last Updated:

    Views: 5954

    Rating: 4.6 / 5 (66 voted)

    Reviews: 81% of readers found this page helpful

    Author information

    Name: Van Hayes

    Birthday: 1994-06-07

    Address: 2004 Kling Rapid, New Destiny, MT 64658-2367

    Phone: +512425013758

    Job: National Farming Director

    Hobby: Reading, Polo, Genealogy, amateur radio, Scouting, Stand-up comedy, Cryptography

    Introduction: My name is Van Hayes, I am a thankful, friendly, smiling, calm, powerful, fine, enthusiastic person who loves writing and wants to share my knowledge and understanding with you.