11 December 2021

Statement

Find the value of the following limit.

\(\lim\limits_{S \to 16} \frac{4-\sqrt{S}}{S-16}\)

Solution

\(\lim\limits_{S \to 16} \frac{4-\sqrt{S}}{S-16}\)

Let's start by removing the square root from the numerator by multiplying by the conjugate expression.

\(= \lim\limits_{S \to 16} \frac{4-\sqrt{S}}{S-16} \times \frac{4+\sqrt{S}}{4+\sqrt{S}} \)

\(= \lim\limits_{S \to 16} \frac{16 - S}{(S-16)(4+\sqrt(S))}\)

\(= \lim\limits_{S \to 16} \frac{16 - S}{S-16} \times \lim\limits_{S \to 16} \frac{16 - S}{4+\sqrt{S}}\)

\(= \lim\limits_{S \to 16} -(\frac{S-16}{S-16}) \times \lim\limits_{S \to 16} \frac{16 - S}{4+\sqrt{S}}\)

\(= -1 \times \frac{1}{8}\)

\(= -\frac{1}{8}\)