Does the Limit exist?

Calculus Level 2

A function $f(x)$ is defined such that $\lim \limits_{x \to a^{+}} f(x) = 3$ , but $\lim \limits_{x \to a^{-}} f(x) = 1$ .

A function $g(x)$ is defined such that $\lim \limits_{x \to a^{+}} g(x) = 1$ , but $\lim \limits_{x \to a^{-}} g(x) = 3$ .

Find whether $\lim \limits_{x \to a} f(x) + g(x)$ exists. And if it does exists, give its value.

It exists, the limit equals

1

It exists, the limit equals

3

It exists, the limit equals

4

It doesn't exist

2 solutions

A Former Brilliant Member
Mar 15, 2021

Remember, a limit exist if both the limit evaluated from negative and positive direction equal each other.

$\lim_{x \rightarrow a^{+}} f(x) + g(x) = \lim_{x \rightarrow a^{+}} f(x) + \lim_{x \rightarrow a^{+}} g(x) = 3 + 1 = 4$

$\lim_{x \rightarrow a^{-}} f(x) + g(x) = \lim_{x \rightarrow a^{-}} f(x) + \lim_{x \rightarrow a^{-}} g(x) = 1 + 3 = 4$

$\lim_{x \rightarrow a^{+}} f(x) + g(x) = \lim_{x \rightarrow a^{-}} f(x) + g(x) = 4$

So the answer is $\boxed{\text{The limit exists and the answer is 4}}$

Jason Gomez
Mar 15, 2021

Examples of $f(x)$ and $g(x)$ are $\frac{|x|}{x} +2$ and $\frac{-|x|}{x} +2$ at $a=0$

Note that the limit exists but the function $f(x)+g(x)$ may not be continuous at a

Yeah, it looks like this on Desmos -

A Former Brilliant Member - 2 months, 4 weeks ago

Add them up now, you will see that at zero it is undefined (obviously) so it isn’t continuous but the limit exists

Then try with sign(x) + 2 and 2- sign(x) ,now it will be continuous

Jason Gomez - 2 months, 4 weeks ago