Answer
Using graphical, numerical and analytic methods, it can be seen that:
$\lim_{x\to16}\dfrac{4-\sqrt{x}}{x-16}=-\dfrac{1}{8}=-0.125$
Work Step by Step
$\lim_{x\to16}\dfrac{4-\sqrt{x}}{x-16}$
From the graph (shown in the answer section), $\lim_{x\to16}\dfrac{4-\sqrt{x}}{x-16}$ can be estimated as $-0.125$
To reinforce this estimation, elaborate a table (shown below), approaching $16$ frmo the left and from the right. It can be also seen that $\lim_{x\to16}\dfrac{4-\sqrt{x}}{x-16}=-0.125$
Let's confirm these estimations by evaluating the limit by analytic methods:
$\lim_{x\to16}\dfrac{4-\sqrt{x}}{x-16}=...$
Factor the denominator as a difference of squares:
$...=\lim_{x\to16}\dfrac{4-\sqrt{x}}{(\sqrt{x}-4)(\sqrt{x}+4)}=...$
Change the sign of the numerator and the sign of the fraction:
$...=\lim_{x\to16}-\dfrac{\sqrt{x}-4}{(\sqrt{x}-4)(\sqrt{x}+4)}=...$
Take the minus sign out of the limit:
$...=-\lim_{x\to16}\dfrac{\sqrt{x}-4}{(\sqrt{x}-4)(\sqrt{x}+4)}=...$
Simplify:
$...=-\lim_{x\to16}\dfrac{1}{\sqrt{x}+4}=...$
Evaluate the limit applying direct substitution:
$...=-\dfrac{1}{\sqrt{16}+4}=-\dfrac{1}{4+4}=-\dfrac{1}{8}$
