Question

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

$\underset{x\to 12}{\lim }\left(\sqrt{x^2+25}-\sqrt{3x}\right)$

Short answer

The value of limit of the given function is$7.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to 12}{\lim }\left(\sqrt{x^2+25}-\sqrt{3x}\right)$

Step 2. Formula used.

The given function is a polynomial, so we can find the limit by direct substitution.

Step 3. Simplification.

Given, $\underset{x\to 12}{\lim }\left(\sqrt{x^2+25}-\sqrt{3x}\right)$

$\begin{array}{c}\underset{x\to 12}{\lim }\left(\sqrt{x^2+25}-\sqrt{3x}\right)=\left(\sqrt{{\left(12\right)}^2+25}-\sqrt{3\times 12}\right)\\ =\sqrt{144+25}-\sqrt{36}\\ =\sqrt{169}-\sqrt{36}\\ =13-6\\ =7\end{array}$

Therefore, our final answer is$7.$