Is it correct to say that $\lim_{x\to\infty}e^x=\infty$? I saw $$\lim_{x\to\infty}e^x=\infty$$ in  a textbook, but I think the limit of the left part doesn't exist. So left part doesn't equal right part. Am I right? 
 A: If you read the lines in your textbook clearly, then you will find there must be somewhere explaining the notation "$= \infty$", which merely stands for, say "gets arbitrarily large", and the symbol $\infty$ does not denote any number.
Usually in calculus we work with the real numbers, and there is no real number called infinity. 
To avoid unnecessary confusion, please be informed that the symbol "$\infty$" is not a number in the sense that we do not allow operations with $\infty$. Certainly you can make $\infty$ a number by stipulating suitable operations with it and suitable notions of "positiveness" and "negativeness" as what measure theory does. 
If you cannot accept the notation that confused you, you may instead write
$$e^{x} \to \infty\ \ as\ \ x \to \infty,$$
though that makes no substantial difference.
A: There's something called the "extended reals," which is $\mathbb R\cup\{-\infty,+\infty\}$ (that is, the real numbers with the addition of infinity and negative infinity). ($\infty-\infty$ is left undefined, as are others.) Limits are usually thought of, at least informally, as being in the extended real number system.
In any case, to be formal, the expression $\lim_{x\to c}f(x)=\infty$ just means that, as $x$ gets closer to $c$, $f(x)$ grows without bound. Technically, the limit doesn't exist (the definition of limit* never mentions infinity), but people write it like that anyway (because, informally, it's thought of as being in the extended reals).
* It's possible that you haven't heard of the formal definition of a limit. It is this:
$\displaystyle\lim_{x\to c}f(x)=L$ means: For all positive numbers $\epsilon$ there is a positive number $\delta$ such that for every $x$ in the interval $(c-\delta,c+\delta)$, $f(x)$ is in the interval $(L-\epsilon,L+\epsilon)$.
$\displaystyle\lim_{x\to\infty}f(x)=L$ means: For all positive numbers $\epsilon$, there is a number $N$ such that for every $x$ greater than $N$, $f(x)$ is in the interval $(L-\epsilon,L+\epsilon)$. (The symbol $\lim\limits_{x\to-\infty}$ is defined similarly.)
Don't worry if you can't wrap your head around those on your first reading. Few people can.
Technically, your limit doesn't exist, because there is no such number $L$. However, people write $\lim_{x\to\infty}e^x=\infty$, because informally, that's what $x$ tends to — infinity. 
A: Note that since $x\to\infty$, it is meaningless to think about the left and right limits because the neighborhood of $\infty$ cannot be approached. We can however use the definition of this limit to show that it is positive, unbounded and ultimately divergent. First note that
$$ \lim\limits_{x\to\infty} f(x) =\infty $$
Is defined by the following statement 
$$ (\forall \epsilon \gt 0)(\exists \delta \gt 0)(x \gt \delta  \Rightarrow f(x) \gt \epsilon) $$
So now let's find a value for $\delta$ that makes the statement true for $f(x)=e^x$.
$$ e^x \gt \epsilon $$
$$ \ln e^x \gt \ln \epsilon $$
$$ x \gt \ln \epsilon $$
So now choose $\delta:= \ln \epsilon$, which implies that
$$ (\forall \epsilon \gt 0)(\exists \delta \gt 0)(x \gt \ln\epsilon \Rightarrow e^x \gt \epsilon) $$
Therefore, by definition 
$$ \lim\limits_{x\to\infty} e^x=\infty $$
