Question

Find the limit (if it exists). $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+5}-\sqrt{5}}{x} $$

Short answer

The limit of the given function as x approaches 0 is \(\frac{1}{2\sqrt{5}}\).

Step-by-step solution

Identify the Problem

First, note that if you substitute x=0 directly into the expression, it results in an indeterminate form, 0/0. This happens due to the radical part, \(\sqrt{x+5}-\sqrt{5}\), turning to zero.

Rationalize the Numerator

We will rationalize the numerator by multiplying by the conjugate of the numerator, \((\sqrt{x+5}+\sqrt{5})\), over itself. Multiply the numerator and denominator by \((\sqrt{x+5}+\sqrt{5})\).\[\lim _{x \rightarrow 0} \frac{\sqrt{x+5}-\sqrt{5}}{x} * \frac{\sqrt{x+5}+\sqrt{5}}{\sqrt{x+5}+\sqrt{5}}\]

Simplify the Expression

Simplify the expression. The numerator simplifies to x, and the denominator simplifies to x(\(\sqrt{x+5}+\sqrt{5}\)). The x in the numerator and the denominator can then be cancelled. \[\lim _{x \rightarrow 0} \frac{x}{x(\sqrt{x+5}+\sqrt{5})}\]Simplify to:\[\lim _{x \rightarrow 0} \frac{1}{\sqrt{x+5}+\sqrt{5}}\]

Substitute x = 0

Now, substitute x = 0 into the equation. The limit value will be:\[ \frac{1}{\sqrt{0+5}+\sqrt{5}} = \frac{1}{2\sqrt{5}} \].