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.

$\underset{x\to 1}{\lim }{x}^{\sin x}$

Short answer

$\underset{x\to 1}{\lim }{x}^{\sin x}=1$

Step-by-step solution

Step 1. Given information

$\underset{x\to 1}{\lim }{x}^{\sin x}$

Step 2. Taking log on both sides

$\begin{array}{r}\underset{x\to 0^{+}}{lim} x^{\sin x}=\underset{x\to 0^{+}}{lim} e^{\ln (x{)}^{\sin x}}\\ =\underset{x\to 0^{+}}{lim} e^{\sin x\ln x}\\ =e^{\underset{x\to 0^{+}}{lim} (\sin x\ln x)}\\ =e^0\end{array}$

Step 3. Calculating the limit

$\underset{x\to 0^{+}}{lim} x^{\sin x}=1$