Algebraic Techniques for Computing Limits

1. The Indeterminate Form: 0/0

When direct substitution into ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ yields $\frac{0}{0}$, it means both the numerator and the denominator are approaching zero simultaneously. You can think of this as a "race to zero." The value of the limit depends on the relative rates at which $f(x)$ and $g(x)$ approach zero. The term "indeterminate" is used because, without further analysis, we cannot determine the outcome. The limit could be a finite number, zero, $+\infty$, $-\infty$, or it might not exist at all.

The following brief reading from Paul's Online Math Notes explains this concept.

Our goal is to algebraically transform the function into an equivalent one that eliminates this ambiguity, allowing us to use direct substitution after all.

2. Technique 1: Factoring and Canceling

This is your primary strategy when dealing with rational functions (a ratio of polynomials) that result in the 0/0 form.

If plugging $x=a$ into a rational function gives 0/0, the Factor Theorem from algebra tells us that $(x-a)$ must be a factor of both the numerator and the denominator. The 0/0 form arises because this common factor is causing a division by zero. Our strategy is to:

1. Factor both the numerator and the denominator.
2. Cancel the common factor $(x-a)$.
3. Evaluate the limit of the new, simplified expression using direct substitution.

This is valid because the original function and the simplified one are identical everywhere except at $x=a$. Since the limit describes the behavior of the function as it approaches $a$, the limits of the two functions are the same.

The following video from The Organic Chemistry Tutor provides several clear examples of this process.

3. Technique 2: Rationalizing with Conjugates

When you encounter a limit with a square root that results in the 0/0 form, factoring is often not an option. This is where rationalization comes in. The key is to multiply both the numerator and the denominator by the conjugate of the expression containing the radical.

The conjugate of an expression like $A-B$ is $A+B$. Multiplying them together leverages the difference of squares formula, $(A-B)(A+B)=A^2-B^2$, which conveniently eliminates the square root.

For instance, to rationalize an expression containing $\sqrt{x}-3$, you would multiply by $\frac{\sqrt{x}+3}{\sqrt{x}+3}$.

The strategy is:

1. Identify the expression containing the radical (e.g., $\sqrt{f(x)}-c$).
2. Multiply the numerator and denominator by its conjugate (e.g., $\sqrt{f(x)}+c$).
3. Simplify the resulting expression. The term that caused the zero in the denominator should now cancel.
4. Evaluate the limit of the simplified expression by direct substitution.

The image below illustrates this process clearly.

Now, let's walk through the steps with a text and video example.

To see this technique in action, the following video provides another excellent walkthrough.

4. Other Algebraic Techniques

Factoring and rationalizing are the two most common methods, but any valid algebraic simplification can be used. For example, when dealing with complex fractions (fractions inside of fractions), a useful first step is often to simplify the expression by multiplying the numerator and denominator by the least common denominator of the "inner" fractions.

The video Calculus 1 - Introduction to Limits demonstrates this technique in the section from handling complex fractions, which you can review if you encounter such problems.

Conclusion

You have now added two essential tools to your calculus toolkit for evaluating limits. When faced with an indeterminate 0/0 form, you have a clear, systematic path forward.

Key Takeaways:

• The 0/0 indeterminate form requires you to algebraically manipulate the function before you can find the limit.
• For rational functions, the primary technique is factoring the numerator and denominator to cancel the term causing the division by zero.
• For functions involving square roots, the technique is rationalization, which involves multiplying the numerator and denominator by the conjugate of the radical expression.
• The overall strategy is to create an equivalent function that is identical to the original near the limit point but is defined at the limit point, allowing for direct substitution.

In our next lesson, we will shift our perspective. Instead of asking what value a function approaches as $x$ nears a specific number, we will investigate the function's "end behavior": what happens to $f(x)$ as $x$ grows infinitely large in the positive or negative direction? This will lead us to the concepts of limits at infinity and horizontal asymptotes.