Question

In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow 4^{-}} \frac{x^{2}}{x^{2}+16} $$

Short answer

The limit of the provided function as \(x\) approaches 4 from the left-hand side is \(\frac{1}{2}\).

Step-by-step solution

Identify the Classification of Limit

From the problem, it can be inferred that the limit should be approached from the left side of the number 4. It is represented as \(x \rightarrow 4^{-}\), which signifies a left-hand limit.

Substitution

Substitute the number 4, which is the limit approaching value, into the original equation. Get: \(\frac{4^{2}}{4^{2}+16}\)

Simplify the Equation

After performing the operations in the numerator and denominator we get: \(\frac{16}{16+16}\) = \(\frac{16}{32}\)

Simplify Further

Finally, simplify \(\frac{16}{32}\) down to its simplest form to get \(\frac{1}{2}\).