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.