Question

Evaluate $\underset{x\to 1}{lim}\frac{f\left(x\right)-f\left(1\right)}{x-1}$ where f(x)= $x^2-2x$

• A. -1
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (B)

0

Given, $f\left(x\right)=x^2-2x$
So, $f\left(1\right)=1^2-2\left(1\right)=-1$
Now, $\underset{x\to 1}{lim}\frac{f\left(x\right)-f\left(1\right)}{x-1}=\underset{x\to 1}{lim}\frac{x^2-2x+1}{x-1}=\underset{x\to 1}{lim}\frac{(x-1)(x-1)}{x-1}=\underset{x\to 1}{lim}(x-1)$
Substituting $x=1$, we get $1-1=0$
Thus, $\underset{x\to 1}{lim}\frac{f\left(x\right)-f\left(1\right)}{x-1}=0$