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Chapter 5: Problem 65

L'Hopital's Rule Determine which of the following limits can be evaluated using L'Hopital's Rule. Explain your reasoning. Do not evaluate the limit. $$\quad\quad\quad\text{(a)}\lim _{x \rightarrow 2} \frac{x-2}{x^{3}-x-6} \quad \text { (b) } \lim _{x \rightarrow 0} \frac{x^{2}-4 x}{2 x-1} $$ $$\text{(c)}\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x}} \quad \text { (d) } \lim _{x \rightarrow 3} \frac{e^{x^{2}}-e^{9}}{x-3}$$ $$\quad\quad\quad\text{(e)}\lim _{x \rightarrow 1} \frac{\cos x}{\ln x} \quad \text { (f) } \lim _{x \rightarrow 1} \frac{1+x(\ln x-1)}{(x-1) \ln x}$$

Short Answer

Expert verified
L'Hopital's Rule can be applied to find the limits in (a), (c), (d), and (f). L'Hopital's Rule can not be applied to find the limits in (b) and (e).

Step by step solution

01

Checking Limit (a)

For lim_(x鈫�2) (x-2)/(x^3 - x - 6), after plugging x = 2, the limit becomes 0/0, which is an indeterminate form. So, L'Hopital's Rule can be applied here.
02

Checking Limit (b)

For lim_(x鈫�0) (x^2 - 4x)/(2x - 1), after plugging x = 0, the limit becomes 0/(-1), which is not an indeterminate form. So, L'Hopital's Rule can not be applied here.
03

Checking Limit (c)

For lim_(x鈫掆垶) x^3/e^x, as x approaches infinity, both the numerator and the denominator approach infinity, so the limit is of the form 鈭�/鈭�, which is an indeterminate form. So, L'Hopital's Rule can be applied here.
04

Checking Limit (d)

For lim_(x鈫�3) (e^(x^2) - e^9)/(x - 3), after plugging x = 3, the limit becomes 0/0, which is an indeterminate form. So, L'Hopital's Rule can be applied here.
05

Checking Limit (e)

For lim_(x鈫�1) cos(x)/ln(x), after plugging x = 1, the limit becomes cos(1)/0, which is not an indeterminate form. So, L'Hopital's Rule can not be applied here.
06

Checking Limit (f)

For lim_(x鈫�1) (1 + x(ln(x) - 1))/((x - 1)ln(x)), after plugging x = 1, the limit is undefined as both the numerator and the denominator approach 0, which is an indeterminate form. So, L'Hopital's Rule can be applied here.

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Key Concepts

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

Indeterminate Forms
Indeterminate forms often arise when evaluating limits in calculus. They occur when the direct substitution of a variable into a limit results in undefined expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These expressions don't have a clear value without further analysis. Recognizing indeterminate forms is crucial because it helps us decide whether advanced techniques, such as L'H么pital's Rule, can be applied to resolve the limit.
For example:
  • In limit (a), \( \lim_{x \rightarrow 2} \frac{x-2}{x^3-x-6} \) yields \( \frac{0}{0} \) when \( x = 2 \). This indicates an indeterminate form.
  • Similarly, limit (c), \( \lim_{x \rightarrow \infty} \frac{x^3}{e^x} \), results in \( \frac{\infty}{\infty} \) as \( x \to \infty \), another indeterminate form.
Understanding these helps in efficiently applying the correct calculus technique.
Limit Evaluation
Evaluating limits involves understanding what happens to a function as the input approaches a certain value or infinity. This process can be straightforward, like plugging values directly into a function. However, indeterminate forms require more careful analysis to resolve.
Indicators to consider when evaluating limits include:
  • The numerator and denominator behavior as they approach a particular point.
  • Identifying if the form results in \( 0/0 \) or \( \infty/\infty \), suggesting the use of L'H么pital's Rule.
When evaluating \( \lim_{x \rightarrow 3} \frac{e^{x^2} - e^9}{x - 3} \), direct substitution gives \( \frac{0}{0} \), indicating a need for further evaluation using L'H么pital's Rule or other techniques. Understanding the properties of the functions involved helps assess the limit effectively.
Calculus Techniques
Calculus provides several techniques to simplify and evaluate complex limits. One popular method is L'H么pital's Rule, which applies to limits presenting indeterminate forms such as \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). This rule involves differentiating the numerator and denominator separately to find the limit.
To apply L'H么pital's Rule successfully:
  • Ensure that the original limit is an indeterminate form.
  • Differentiation of the numerator and denominator must be done independently.
  • If applying once doesn't resolve the indeterminate form, it can be repeated as needed.
For instance, in limit (f), \( \lim_{x \rightarrow 1} \frac{1 + x(\ln x - 1)}{(x-1)\ln x} \), the form \( \frac{0}{0} \) allows L'H么pital's Rule to find a more straightforward result. Techniques like this aid in smoothly handling complex calculus problems.

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Most popular questions from this chapter

Verifying Identities In Exercises 81 and \(82,\) verify each identity. (a) \(\operatorname{arccsc} x=\arcsin \frac{1}{x}, \quad|x| \geq 1\) (b) \(\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}, \quad x>0\)

In Exercises 57-60, use a graphing utility to graph the slope field for the differential equation and graph the particular solution satisfying the specified initial condition. $$\begin{array}{l}{\frac{d y}{d x}=\frac{1}{12+x^{2}}} \\\ {y(4)=2}\end{array}$$

Modeling Data A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate \(R\) (in liters per hour) at time \(t\) (in hours) is given in the table. \(\begin{array}{|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} \\ \hline R & {425} & {240} & {118} & {71} & {36} \\ \hline\end{array}\) (a) Use the regression capabilities of a graphing utility to find a linear model for the points \((t, \ln R) .\) Write the resulting equation of the form \(\ln R=a t+b\) in exponential form. (b) Use a graphing utility to plot the data and graph the exponential model. (c) Use a definite integral to approximate the number of liters of chemical released during the 4 hours.

Evaluating a Definite Integral In Exercises \(77-80\) , evaluate the definite integral. Use a graphing utility to verify your result. $$\int_{-4}^{4} 3^{x / 4} d x$$

Calculus History InL'Hopital's 1696 calculus textbook, he illustrated his rule using the limit of the function $$f(x)=\frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}}$$ as \(x\) approaches \(a, a>0 .\) Find this limit.
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