Question

Find \(\lim_{x\rightarrow 0}\) f(x) and \(\lim_{x\rightarrow 1}\) f(x) where { 2x+3, x≤0 3 (x+1), x>0

Answer 1

The given function is
f(x) = { 2x+3, x≤0 3(x+1) x>0
\(\lim_{x\rightarrow 0^-}\) f(x) = \(\lim_{x\rightarrow 0}\) [2x+3] =  (2(0)+3=3
\(\lim_{x\rightarrow 0^-}\) f(x) = \(\lim_{x\rightarrow 0}\) 3(x+1)  =  3(0+1)= 3
∴ \(\lim_{x\rightarrow 0^-}\) f(x) = \(\lim_{x\rightarrow 0^+}\) f(x) = \(\lim_{x\rightarrow 0}\) f(x) = 3
\(\lim_{x\rightarrow 1^-}\) f(x) = \(\lim_{x\rightarrow 0^+}\) 3(x+1) =  3(1+1) =6
\(\lim_{x\rightarrow 1^-}\) f(x) = \(\lim_{x\rightarrow 1}\) 3(x+1)  = 3(1+1) =6
∴ \(\lim_{x\rightarrow 1^-}\) f(x) = \(\lim_{x\rightarrow 1^+}\) f(x) = \(\lim_{x\rightarrow 1}\)f(x) = 6