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I am aware that the definition of a limit $L$ for an infinite sequence $a_n$ is defined as follows

A real sequence $a_n$ tends to a limit $L$ as $n \rightarrow \infty$ if, for any arbitraty $\varepsilon > 0$, there exists $N \in \mathbb{Z}^+$ such that, for all $n \geq N$ we have $$ | a_n - L | < \varepsilon $$

However, I am unsure of how we would prove that a function $f(x)$ tends to some limit $L^*$ as $x \rightarrow x_0$.

Having thought about it, I've decided that it may be done by finding some sequence $x_n$ that tends to $x_0$ as $n \rightarrow \infty$, substituting this sequence into the function to give a new sequence $a_n = f(x_n)$.

Would proving that $a_n \rightarrow L^*$ as $n \rightarrow \infty$ suffice to show that the function converges to the limit $L^*$?

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    You need to prove that $a_n \to x_0$ implies $f(a_n) \to L^*$ for all sequences that converge to $x_0$. – 2017-02-09
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    This is correct (so long as you do it for all sequences). – 2017-02-09
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    I can see clearly how it would be done for a particular sequence that converges to $x_0$, but how would you show it holds for all sequences that converge to $x_0$? – 2017-02-09

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If a sequence $\{a_k\}$ such that $a_k\rightarrow x$ as $a_k\rightarrow\infty$ has the property that $f(a_k)\rightarrow L$ you can not conclude that $f(x)\rightarrow L$. You can say at most that if $f(x)$ has a limit it most be $L$

There is a theorem that may clarify this

$f(x)\rightarrow L$ as $x\rightarrow x_0$ if and only if for every sequence $\{a_k\}$ such that $a_k\rightarrow x_0$ we have that $f(a_k)\rightarrow L$.

That is you have to prove that the image of every sequence that tends to $x_0$ converges to the same number. In general this is very impractical. This theorem is important in its contrapositive form:

If there exist some sequence sequence $\{a_k\}$ such that $a_k\rightarrow x_0$ but $f(a_k)\not\rightarrow L$ then $f(x)\not\rightarrow L$ as $x\rightarrow x_0$.

This is very useful to disprove that a function has limit, see for example: $$f(x)=\frac{|x|}{x}$$ and you want to know how this function behave as $x\rightarrow 0$ (observe that it is not define in this point). Try the sequences $a_k=\frac{1}{n}$ and $b_k=-\frac{1}{n}$ to find that this function do not have any limit as we approach to zero.

There is a list of useful theorems to deal with limits. As you get more advanced in your calculus course you will find then useful and familiar. If this is your first encounter with a limit in a formal try to study carefully the proves by definition that something is a limit, it is extremely important to understand how $\epsilon-\delta$ behaves.

Hope you find this useful.