Q. 69 - Calculus

Question

In Exercises 69-80, determine whether or not $f$ is continuous and/or differentiable at the given value of $x$. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

$f\left(x\right)=\frac{1}{x},x=0$

Step-by-Step Solution

Answer

The function is non-continuous and non-differentiable at $x=0$ and graph is 

Step 1. Given information

Given function $f\left(x\right)=\frac{1}{x},x=0$

Step 2: Calculate the LHL and RHL and calculate

Calculating, we get

$$\begin{gathered} \underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{-}}{\lim }\frac{1}{x} \\ \underset{x\to 0^{-}}{\lim }f\left(x\right)=\frac{1}{0} \\ =undefined \end{gathered}$$

Similarly

$$\begin{gathered} \underset{x\to 0^{+}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }\frac{1}{x} \\ \underset{x\to 0^{+}}{\lim }f\left(x\right)=\frac{1}{0} \\ =undefined \end{gathered}$$

$$\begin{gathered} f\left(0\right)=\frac{1}{0} \\ =undefined \end{gathered}$$

So the function is not continuous at $x=0$

Step 3: Checking for differentiability

Checking, we get

$$\begin{gathered} \underset{x\to 0^{-}}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0}=\underset{x\to 0^{-}}{\lim }\frac{\frac{1}{x}-\frac{1}{0}}{x-0} \\ =undefined \\ \underset{x\to 0^{+}}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0}=\underset{x\to 0^{+}}{\lim }\frac{\frac{1}{x}-\frac{1}{0}}{x-0} \\ =undefined \end{gathered}$$

The function is not differentiable at $x=0$

Step 4: Sketching the graph

The graph is