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Chapter 4: Problem 15

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. \(\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}}\)

Short Answer

Expert verified
The limit is \(\frac{1}{2}\).

Step by step solution

01

Check the Indeterminate Form

First, we'll substitute \(x = 0\) into the expression \(\frac{e^x - 1 - x}{x^2}\). After substitution, we get \(\frac{e^0 - 1 - 0}{0^2} = \frac{0}{0}\). This is an indeterminate form, so l'Hospital's Rule can be applied.
02

Apply l'Hospital's Rule

Since the limit results in an indeterminate form \(\frac{0}{0}\), we apply l'Hospital's Rule, which requires us to differentiate the numerator and the denominator. The derivative of the numerator \(e^x - 1 - x\) is \(e^x - 1\), and the derivative of the denominator \(x^2\) is \(2x\). Thus, the limit becomes \(\lim_{x \to 0} \frac{e^x - 1}{2x}\).
03

Re-check the Indeterminate Form

Substitute \(x = 0\) in the new expression \(\lim_{x \to 0} \frac{e^x - 1}{2x}\). It again becomes \(\frac{0}{0}\), which is indeterminate. This allows us to apply l'Hospital's Rule again.
04

Apply l'Hospital's Rule Again

Differentiate the numerator \(e^x - 1\) to get \(e^x\) and the denominator \(2x\) to get \(2\). The new limit to evaluate is \(\lim_{x \to 0} \frac{e^x}{2}\).
05

Evaluate the Limit

Substitute \(x = 0\) into the simplified expression \(\frac{e^x}{2}\), which gives \(\frac{e^0}{2} = \frac{1}{2}\). This limit exists and is the result of the original limit problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

l'Hospital's Rule
When you encounter a limit problem that results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), l'Hospital's Rule is a powerful technique to find the limit. This rule allows you to differentiate the numerator and the denominator separately and then take the limit again.

The key steps are:
  • Identify the indeterminate form \( \frac{f(x)}{g(x)} \).
  • Differentiate the numerator to get \( f'(x) \).
  • Differentiate the denominator to get \( g'(x) \).
  • Calculate the limit of \( \frac{f'(x)}{g'(x)} \) as \( x \) approaches the given point.

It's important to check if the result is still indeterminate after the first application. Sometimes, you may need to apply the rule more than once, as seen in the original exercise.
Indeterminate Forms
In calculus, indeterminate forms occur when substituting a certain value into a limit expression results in an undefined form. The most common indeterminate forms include \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \).

These forms tell us that the limit could potentially have many different values, making it necessary to use special techniques to resolve them. Some techniques include:
  • Using algebraic manipulation to simplify the expression.
  • Applying calculus rules such as l'Hospital's Rule.
  • Rewriting in a format that reveals the value of the limit.

The goal is to remove the indeterminacy and find a concrete limit value.
Differentiation
Differentiation is the process of finding the derivative of a function. It provides a way to calculate the rate at which a function changes.

For instance, in the original exercise, the differentiation of the function \( e^x - 1 - x \) with respect to \( x \) yields \( e^x - 1 \), and the differentiation of \( x^2 \) yields \( 2x \).

Why is this useful? Differentiation helps in simplifying limits, especially when using l'Hospital's Rule, because it allows us to replace complicated rate functions with simpler derivatives. It is a fundamental tool in calculus for analyzing the behavior and changes within any system.
Exponentials
Exponential functions, like \( e^x \), are functions with constant bases raised to variable exponents. They are prevalent in calculus due to their unique properties. One such property is that the derivative of \( e^x \) is precisely itself, \( e^x \).

This feature simplifies analysis in many calculus problems. In the given exercise, the presence of \( e^x \) shows this characteristic well, as its differentiation smoothly leads to a simpler form, allowing easy evaluation of the limit.
  • Exponential growth functions describe rapidly increasing rates in applications such as population growth.
  • They always remain positive, which can help in ensuring certain conditions within calculus problems.
Understanding exponential functions' behavior is critical for solving limits and many other calculus-based applications.

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Most popular questions from this chapter

Find the most general antiderivative of the function.(Check your answer by differentiation.) \(c(t)=\frac{3}{t^{2}}, t>0\)

$$ \begin{array}{c}{\text { The family of bell-shaped curves }} \\\ {y=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / / 2 \sigma^{2})}}\end{array} $$ $$ \begin{array}{l}{\text { occurs in probability and statistics, where it is called the }} \\ {\text { normal density function. The constant } \mu \text { is called the mean }} \\ {\text { and the positive constant } \sigma \text { is called the standard deviation. }}\end{array} $$ $$ \begin{array}{c}{\text { For simplicity, let's scale the function so as to remove the }} \\ {\text { factor } 1 /(\sigma \sqrt{2 \pi}) \text { and let's analyze the special case where }} \\ {\mu=0 . \text { So we study the function }} \\ {f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)}}\end{array} $$ $$ \begin{array}{l}{\text { (a) Find the asymptote, maximum value, and inflection }} \\ {\text { points of } f .} \\ {\text { (b) What role does } \sigma \text { play in the shape of the curve? }} \\ {\text { (c) Illustrate by graphing four members of this family on }} \\ {\text { the same screen. }}\end{array} $$

Find the most general antiderivative of the function.(Check your answer by differentiation.) \(f(x)=2 x+3 x^{1.7}\)

Find the point on the curve \(y=\sqrt{x}\) that is closest to the point \((3,0) .\)

A cubic function is a polynomial of degree \(3 ;\) that is, it has the form \(f(x)=a x^{3}+b x^{2}+c x+d,\) where \(a \neq 0\) . (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities. (b) How many local extreme values can a cubic function have?
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