Q. 20 - Calculus

Question

Use a calculator or other graphing utility to investigate the graph $f\left(x\right)=x\sin \left(\frac{1}{x}\right)$ near x = 0. Be sure to have your calculator set to radian mode. Use the graphs to make an educated guess for $\underset{x\to 0}{lim} f\left(x\right)$.

Step-by-Step Solution

Answer

f(x) is undefined near x=0

Step 1. Given information.

We have been given a function $f\left(x\right)=x\sin \left(\frac{1}{x}\right)$.

We have to use a calculator or other graphing utility to investigate the graph of the function near x = 0.

Step 2. Graph the function

Function is :

$f\left(x\right)=x\sin \left(\frac{1}{x}\right)$

The graph is :

We can see that at x=0 is not defined.

Step 3. Solving the limit expression.

Solving the limit expression,

$\begin{array}{r}\underset{x\to 0}{lim} f\left(x\right)=\underset{x\to 0}{lim} x\sin \left(\frac{1}{x}\right)\\ =0\sin \left(\frac{1}{0}\right)\\ =0\sin \left(\mathrm{\infty }\right)\end{array}$

                        = undefined