Question

Calculate each of the limits in Exercises 49–64. Some of these limits are made easier by considering the logarithm of the limit first, and some are not.

$\underset{x\to 0}{\lim }{(\cos x)}^{\frac{1}{x}}$

Short answer

$\underset{x\to 0}{\lim }{(\cos x)}^{\frac{1}{x}}=1$

Step-by-step solution

Step 1. Given information

$\underset{x\to 0}{\lim }{(\cos x)}^{\frac{1}{x}}$

Step 2. Taking log on both sides

$$\begin{gathered} \begin{array}{r}\underset{x\to 0^{+}}{lim} \ln (\cos x{)}^{\frac{1}{x}}=\underset{x\to 0^{+}}{lim} \frac{1}{x}\ln (\cos x)\\ =\underset{x\to 0^{+}}{lim} \frac{\ln (\cos x)}{x}\\ =\underset{x\to 0^{+}}{lim} \frac{1}{\cos x}\\ =\underset{x\to 0^{+}}{lim} \frac{1}{\cos x}\end{array} \\ =1 \end{gathered}$$

Step 3. Calculating the limit

$\begin{array}{r}\underset{x\to 0^{+}}{lim} (\cos x{)}^{\frac{1}{x}}=e^1\\ =e\end{array}$