Question

how do you find the limit of the following equation as x approaches 0, please outline all steps involved including the factorization

Answer 1

Given:

`\lim_(x\to0)((√(x+1)-1)/(x))`

To solve, we have

step 1: From rationalization.

This gives:

`\begin{gathered} (1)/(√(x+1)+1)*(√(x+1)-1)/(√(x+1)-1) \\ =(√(x+1)-1)/((√(x+1))^2-1) \\ =(√(x+1)-1)/(x+1-1) \\ \implies(√(x+1)-1)/(x) \\ This\text{ implies that} \\ (√(x+1)-1)/(x)\text{ is equaivalent to}(1)/(√(x+1)+1) \end{gathered}`

step 2: Substitute the value of zero for x.

Thus, we have

`\begin{gathered} \lim_(x\to0)((1)/(√(x+1)+1)) \\ when\text{ x =0, we have} \\ (1)/(√(0+1)+1)=(1)/(2) \end{gathered}`

Hence, we have

`\lim_(x\to0)((√(x+1)-1)/(x))=(1)/(2)`