graph limit problem

Question

For this given graph, since it approaches positive infinity from the left and right sides of $x=2$, shouldn't $\lim_{x \to 2} g(x) = \infty$? Or would its value not exist (as it says in my homework answer key)?

Answer 1

From the graph,

$$\lim_{x\to 2, x <2}g (x)=+\infty $$

and

$$\lim_{x\to 2, x>2}g (x)=+\infty $$

thus $$\lim_{x\to 2, x\ne 2}g (x )=+\infty $$

Answer 2

If you are looking for a value $\ell\in\mathbb{R}$ as a solution of this limit, then the limit would not exist. Remember that the definition of limit for a real valued function says as follows:

Suppose $f:\mathbb{R}\to\mathbb{R}$ is defined on the real line and $p,\ell\in\mathbb{R}$. It is said the limit of $f$, as $x$ approaches $p$, is $\ell$ and written $$\lim_{x\to p} f(x)=\ell,$$ if the following property holds: for every real $\varepsilon > 0$, there exists a real $\delta > 0$ such that for all real $x$, $0 < | x − p | < \delta$ implies $| f(x) − \ell | < \varepsilon$.

But limits can also have infinite values. Sometimes, rather than say that a limit is infinity, the proper thing is to say that the function diverges or grows without bound. In any case, the limit of $f$ as $x$ approaches $a$ is infinity, denoted $$\lim _{x\to a}f(x)=\infty,$$ means that for all $\varepsilon > 0$ there exists $\delta >0$ such that $f(x)>\varepsilon$ whenever $|x-a|<\delta$.

Deciding what to answer in your homework is up to you. You should review your notes and check how your professor defined limits.