Q. 5 - Calculus

Question

Explain why we can’t calculate every limit $\underset{x\to c}{\lim }f\left(x\right)$ just by evaluating f(x) at $x = c$. Support your argument with the graph of a function f for which $\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right)$.

Step-by-Step Solution

Answer

We cannot calculate every limit $\underset{x\to c}{\lim }f\left(x\right)$ just by evaluating f(x) at $x=c$.

Step 1. Given information.

We cannot calculate every limit $\underset{x\to c}{\lim }f\left(x\right)$ just by evaluating $f\left(x\right)$ at $x=c$.

Step 2. Explanation.

For the limit expression $\underset{x\to 1}{\lim }\frac{x-1}{x^2-1}$, we cannot put $x=1$ in the limit expression to get the value.

$$\begin{gathered} \underset{x\to 1}{\lim }\frac{x-1}{x^2-1} \\ =\frac{1-1}{1^2-1} \\ =\frac{0}{1-1} \\ =\frac{0}{0} \end{gathered}$$

So, $\frac{0}{0}$ is undefined.