Question

Evaluate $\lim_{x\rightarrow 5}$ f(x), where f(x) = |x|-5

Answer 1

The given function is f(x) = |x|-5
$\lim_{x\rightarrow 5^-}$ f(x) = $\lim_{x\rightarrow 5^-}$ [|x|-5]
= $\lim_{x\rightarrow 5}$(x-5) [When x>0, |x| = x]
= 5-5
= 0
$\lim_{x\rightarrow 5^+}$f(x) =$\lim_{x\rightarrow 5^+}$ [|x|-5|]
= $\lim_{x\rightarrow 5}$ (x-5) [When x > 0, |x| = x]
= 5-5
= 0
It is observed that $\lim_{x\rightarrow 5^-}$ f(x) =$\lim_{x\rightarrow 5^+}$f(x)
Hence,$\lim_{x\rightarrow 5}$ = 5.