Let f(x) = {x+5,if x>0 x-4,if x < 0. Prove that lim f(x), x ∈ 0.

Question

Let \(f(x) = \begin{cases} x+5, & \quad \text{if } x >{0}\\ x-4, & \quad \text{if } x<{ 0} \end{cases} \). Prove that \(\lim\limits_{x \to 0}\) f(x) does not exist.

f(x) = {x+5,if x>0 x-4,if x < 0

lim f(x), x ∈ 0

Answer 1

Given,

f(x) = {x+5,if x>0 x-4,if x < 0

To find whether lim f(x), x ∈ 0 exists?

To limit to exist we know,

Thus to limit to exist,

From above equations,

Thus, 

The limit \(\lim\limits_{x \to 0}\) f(x) does not exists.