Q. 57 - Calculus

Question

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47–60.$

$\underset{x\to 1}{\lim }\frac{x^2-1}{x-1}=2$

Step-by-Step Solution

Answer

Delta-epsilon proof is, 

Whenever $0<|x-1|<\delta$, we also have $\left|\frac{x^2-1}{x-1}-2\right|<ϵ$. 

Step 1. Given information

We are given,

$\underset{x\to 1}{\lim }\frac{x^2-1}{x-1}=2$

Step 2. Writing the delta-epsilon proofs

The strategy is to write delta-epsilon proofs for the given limit statement.

Consider that $\in >0$, choose $\delta =ϵ$.

For all x with $0<|x-1|<\delta$, we also have $\left|\frac{x^2-1}{x-1}-2\right|<ϵ$, so

$$\begin{gathered} =\left|\frac{(x-1)(x+1)}{(x-1)}-2\right| \\ =|x+1-2| \\ =|x-1| \\ <\delta \\ =\in \end{gathered}$$

So, whenever $0<|x-1|<\delta$, we also have $\left|\frac{x^2-1}{x-1}-2\right|<ϵ$.