Introduction

In calculus, we often need to find the limit of a function as it approaches a certain value. A common scenario involves the limit of a ratio of two functions, which we call rational functions. The general form looks like this:

$$\lim_{x \to a} \frac{f(x)}{g(x)}$$

This expression asks: “What value does the fraction approach as $x$ gets closer and closer to $a$?”

Direct Substitution: The First Approach

The simplest way to find a limit is by direct substitution. This means we simply plug in $x = a$ into both $f(x)$ and $g(x)$ and see what we get.

Example:

Consider the limit:

$$\lim_{x \to 4} \frac{x^2 – 16}{x – 4}$$

Let’s try direct substitution. Plugging in $x = 4$ gives us:

$$\frac{4^2 – 16}{4 – 4} = \frac{16 – 16}{0} = \frac{0}{0}$$

Uh-oh! We got $\frac{0}{0}$, which is an indeterminate form. This means we can’t find the limit just by plugging in the value. But don’t worry—this doesn’t mean the limit doesn’t exist. We just need a different approach.

Factoring: A Useful Technique

When direct substitution fails, especially when you get $\frac{0}{0}$, the next step is often to try factoring.

Example (continued):

Let’s go back to our example:

$$\lim_{x \to 4} \frac{x^2 – 16}{x – 4}$$

Notice that the numerator $x^2 – 16$ is a difference of squares, which can be factored as:

$$x^2 – 16 = (x + 4)(x – 4)$$

So, we can rewrite the limit as:

$$\lim_{x \to 4} \frac{(x + 4)(x – 4)}{x – 4}$$

Now, we see that $(x – 4)$ appears in both the numerator and the denominator. We can cancel it out (as long as $x \neq 4$), leaving us with:

$$\lim_{x \to 4} (x + 4)$$

Now, direct substitution works:

$$4 + 4 = 8$$

So, the limit is $8$.

L’Hôpital’s Rule: Another Powerful Tool

Sometimes, factoring isn’t enough, or it’s too tricky. In such cases, we can use L’Hôpital’s Rule. This rule is very handy when direct substitution gives you $\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$.

L’Hôpital’s Rule: If you get an indeterminate form, you can take the derivative of the numerator and the derivative of the denominator separately, and then try substitution again.

L’Hôpital’s Rule in Action

Let’s use L’Hôpital’s Rule on our example:

$$\lim_{x \to 4} \frac{x^2 – 16}{x – 4}$$

We already know that direct substitution gives us $\frac{0}{0}$, so L’Hôpital’s Rule applies.

First, find the derivatives:

• Derivative of the numerator $x^2 – 16$ is $2x$.
• Derivative of the denominator $x – 4$ is $1$.

Now, apply L’Hôpital’s Rule:

$$\lim_{x \to 4} \frac{2x}{1}$$

Substitute $x = 4$:

$$\frac{2 \cdot 4}{1} = \frac{8}{1} = 8$$

So, once again, the limit is $8$.

Conclusion

When you’re faced with a limit of a rational function, start with direct substitution. If that gives you an indeterminate form like $\frac{0}{0}$, try factoring or use L’Hôpital’s Rule to find the limit. With practice, you’ll get better at spotting which technique to use.