Answer:
[tex] \displaystyle 8[/tex]
Step-by-step explanation:
we would like to compute the following limit
[tex] \displaystyle \lim_{x \to 16} \left( \frac{x - 16}{ \sqrt{x} - 4} \right) [/tex]
if we substitute 16 directly we'd end up
[tex] \displaystyle = \frac{16 - 16}{ \sqrt{16} - 4} [/tex]
[tex] \displaystyle = \frac{0}{ 0} [/tex]
which isn't a good answer now notice that we have a square root on the denominator so we can rationalise the denominator to do so multiply the expression by √x+4/√x+4 which yields:
[tex] \displaystyle \lim_{x \to 16} \left( \frac{x - 16}{ \sqrt{x} - 4} \times \frac{ \sqrt{x} + 4 }{ \sqrt{x} + 4 } \right) [/tex]
simplify which yields:
[tex] \displaystyle \lim_{x \to 16} \left( \frac{(x - 16)( \sqrt{x} + 4)}{ x - 16} \right) [/tex]
we can reduce fraction so that yields:
[tex] \displaystyle \lim_{x \to 16} \left( \frac{ \cancel{(x - 16)}( \sqrt{x} + 4)}{ \cancel{x - 16} } \right) [/tex]
[tex] \displaystyle \lim _{x \to 16} \left( \sqrt{x } + 4\right) [/tex]
now it's safe enough to substitute 16 thus
substitute:
[tex] \displaystyle = \sqrt{16} + 4[/tex]
simplify square root:
[tex] \displaystyle = 4 + 4[/tex]
simplify addition:
[tex] \displaystyle = 8[/tex]
hence,
[tex] \displaystyle \lim_{x \to 16} \left( \frac{x - 16}{ \sqrt{x} - 4} \right) = 8[/tex]
