Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 5^{+}} \frac{x-5}{x^{2}-25} $$

Short answer

The limit of given function as \(x\) approaches \(5^{+}\) is \( \frac{1}{10} \).

Step-by-step solution

Simplify expression

Firstly, simplify the expression in question by factoring the terms in the equation. In denominator, \(x^{2} - 25\) can be expressed as \((x - 5)(x+5)\) by applying the difference of squares formula. Thus, the equation becomes: \(\frac{x-5}{(x - 5)(x + 5)}\)

Cancel similar terms

After simplifying, notice that \(x - 5\) on the top and bottom of the expression cancels out, leaving behind \(\frac{1}{x + 5}\). Therefore, the expression can be written as \(\lim _{x \rightarrow 5^{+}} \frac{1}{x + 5}\)

Compute limit

Finally, substitute '5' for all occurrences of \(x\) in the equation. Doing so, we get \(\frac{1}{5 + 5}\) which gives us \(\frac{1}{10}\).