Question

Evaluate the limit and justify each step by indicating the appropriate limit law(s).$\underset{x\to 2}{\lim }\frac{x^2+x-6}{x-2}$

Short answer

The value of limit of the given function is$5.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,$\underset{x\to 2}{\lim }\frac{x^2+x-6}{x-2}$

Step 2. Formula used.

The given function is a rational number. We can find the limit by cancelling a common factor.

Step 3. Simplification.

Given,$\underset{x\to 2}{\lim }\frac{x^2+x-6}{x-2}$

$\begin{array}{c}\underset{x\to 2}{\lim }\frac{x^2+x-6}{x-2}=\frac{x^2+(3-2)x-6}{x-2}\\ =\frac{x^2+3x-2x-6}{x-2}\\ =\frac{x(x+3)-2(x+3)}{x-2}\\ =\frac{(x+3)(x-2)}{x-2}\\ =x+3\\ \end{array}$

Now, we apply the value:

$\begin{array}{c}=2+3\\ =5\end{array}$

Therefore, our final answer is 5.