Q. 30 - Calculus

Question

Consider the limit values below:

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=3,\underset{x\to 2^{+}}{\lim }f\left(x\right)=3\text{ and }f\left(2\right)=0$

The strategy is to sketch the graph of the function having the above limit values

Step-by-Step Solution

Answer

Step 1. Given

The given limit values are

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=3,\underset{x\to 2^{+}}{\lim }f\left(x\right)=3\text{ and }f\left(2\right)=0$

Step 2. Breakup point

The left-hand limit,$\underset{x->2^{-}}{\lim }f\left(x\right)=3$ represents that the function value approaches to 3 as x  approaches to 2 from the left side and the right-hand limit, $\underset{x->2^{+}}{\lim }f\left(x\right)=3$ represents that the function value approaches to 3 as x approaches to 2 from the right side.

Thus, the limit of the function at x = 2is $\underset{x->2}{\lim }f\left(x\right)=3$

But the value of the function at x = 2 is given 0. That is, f(2) = 0 .

This means that the function value is not same as limit value. This implies that the function is not continuous at x = 2 because $\underset{x->2^{-}}{\lim }f\left(x\right)\varnothing f\left(2\right)$

Hence, the function is discontinuous at x = 2 . So, there is a break in the graph at point

x = 2

Comment

Step 3. Graph obtained