Question

Evaluate the limit and justify each step by indicating the appropriate limit law(s).

$\underset{x\to -1}{\lim }\frac{x-2}{x^2+4x-3}$

Short answer

The value of limit of the given function is$\frac{1}{2}$.

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to -1}{\lim }\frac{x-2}{x^2+4x-3}$

Step 2. Formula used.

The given function is a rational number and$x$ is in its domain. So we can find the limit by direct substitution.

Step 3. Simplification.

Given, $\underset{x\to -1}{\lim }\frac{x-2}{x^2+4x-3}$

Put the value by direct substitution.

$\underset{x\to -1}{\lim }\frac{x-2}{x^2+4x-3}=\frac{-1-2}{\begin{array}{c}\text{Invalid <m:msup> element}+4\times (-1)-3\\ =\frac{-3}{1-4-3}\\ =\frac{-3}{\begin{array}{l}-6\\ =\frac{1}{2}\end{array}}\end{array}}$

Therefore, our final answer is$\frac{1}{2}.$