Question

Each of the limits in Exercises 7–12 is of the indeterminate form $0·\infty$or $\infty ·0$. Rewrite each limit so that it is (a) in the form $\frac{0}{0}$and then (b) in the form $\frac{\infty }{\infty }$. Then (c) determine which of these indeterminate forms would be easier to work with when applying L’Hopital’s rule.

11.role="math" localid="1648641810672" $\underset{x\to 0}{\lim }x\csc x$

Short answer

Part (a) The limit in the form $\frac{0}{0}$is $\underset{x\to 0}{\lim }\frac{x}{\sin x}$

Part (b) The limit in the form $\frac{\infty }{\infty }$is $\underset{x\to 0}{\lim }\frac{\csc x}{{\displaystyle \frac{1}{x}}}$

Part (c) Part (a) is easier to work with L'Hospital's rule.

Step-by-step solution

Part (a) Step 1. Given information

We have been given the limit$\underset{x\to 0}{\lim }x\csc x$

Part (a) Step 2. Write the limit in the form 00

$$\begin{gathered} \underset{x\to 0}{\lim }x\csc x \\ =\underset{x\to 0}{\lim }\frac{x}{\sin x} \end{gathered}$$

Part (b) Step 1. Write the limit in the form ∞∞

$$\begin{gathered} \underset{x\to 0}{\lim }x\csc x \\ =\underset{x\to 0}{\lim }\frac{x}{{\displaystyle \frac{1}{x}}} \end{gathered}$$

Part (c) Step 1. Determine which of these indeterminate forms would be easier to work with when applying L’Hopital’s rule.

$$\begin{gathered} \underset{x\to 0}{\lim }x\csc x \\ =\underset{x\to 0}{\lim }\frac{x}{\sin x} \\ =\underset{x\to 0}{\lim }\frac{1}{\cos x} \\ =\underset{x\to 0}{\lim }\frac{1}{\cos 0} \\ =1 \end{gathered}$$

Therefore, Part (a) is easier to work with L'Hospital's rule.