Question

Find the limit (if it exists). $$ \lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25} $$

Short answer

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

Step-by-step solution

Simplify The Function

First, try to simplify the function. Notice that the denominator is a difference of squares \( a^{2}-b^{2}=(a-b)(a+b) \). Apply that rule to simplify the fraction: \( \frac{x-5}{(x-5)(x+5)} \).

Cancel Common Terms

Observe that \( x - 5 \) is a common term in the numerator and denominator, therefore cancel this term. This simplifies the expression to \( \frac{1}{x+5} \).

Direct Substitution

With the expression simplified, we can substitute the limiting value of \( x = 5 \) into the function \( \frac{1}{x+5} \) to yield \( \frac{1}{5+5} \).