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