Algebraic Techniques for Evaluating Limits

1. The Indeterminate Form: $\frac{0}{0}$

When direct substitution gives you $\frac{0}{0}$, this is known as an indeterminate form. It's crucial to understand that this is not the answer. It doesn't mean the limit is 0, 1, or that it doesn't exist. It simply means the method of direct substitution failed, and we need a more clever approach.

Think of it as a "race to zero" between the numerator and the denominator. The final value of the limit depends on how the two expressions approach zero relative to each other. To find the answer, we must simplify the function to resolve this ambiguity.

The following reading provides a formal introduction to this concept.

2. Technique 1: Factorising and Cancelling

This is your primary tool for dealing with indeterminate forms in rational functions (one polynomial divided by another).

The logic is rooted in the Factor Theorem, which you've encountered before. If substituting $x=a$ into a polynomial gives you zero, then $(x-a)$ must be a factor of that polynomial. So, if both the numerator and denominator of a rational function become zero at $x=a$, they must share a common factor of $(x-a)$. This common factor is the source of our $\frac{0}{0}$ problem.

Our strategy is:

1. Factorise the numerator and denominator.
2. Cancel the common factor $(x-a)$.
3. Evaluate the limit of the simplified function using direct substitution.

This works because the simplified function is identical to the original at every point except at $x=a$, and the limit doesn't care about the value precisely at that point.

The following video provides excellent examples of this technique.

Example: Find ${\lim }_{x\to 4}\frac{x^2-16}{x-4}$.

1. Direct Substitution: $\frac{4^2-16}{4-4}=\frac{16-16}{4-4}=\frac{0}{0}$. Indeterminate form.
2. Factorise: The numerator is a difference of two squares, $x^2-4^2$, which factors to $(x-4)(x+4)$.
   ${\lim }_{x\to 4}\frac{(x-4)(x+4)}{x-4}$
3. Cancel: We cancel the common factor $(x-4)$.
   ${\lim }_{x\to 4}(x+4)$
4. Evaluate: Now, use direct substitution on the simplified expression.
   $4+4=8$
   So, ${\lim }_{x\to 4}\frac{x^2-16}{x-4}=8$.

As a quick refresher, here are some common factorisation patterns.

3. Technique 2: Rationalising with the Conjugate

When you encounter an indeterminate form involving a square root, factorising is often not an option. Instead, we use a technique called rationalising. This involves multiplying the numerator and denominator by the conjugate of the expression containing the root.

The conjugate of an expression like $\sqrt{a}-b$ is $\sqrt{a}+b$. The magic of this is that when you multiply them, you get $(\sqrt{a}{)}^2-b^2=a-b^2$, which eliminates the square root.

The strategy is:

1. Identify the expression with the square root that is causing the zero (e.g., $\sqrt{x+5}-3$).
2. Multiply both the numerator and the denominator by its conjugate (e.g., $\sqrt{x+5}+3$).
3. Simplify the expression. A common factor should appear.
4. Cancel the common factor and evaluate the new limit by direct substitution.

Let's watch this technique in action.

Here is a static image showing a complete worked example of this process.

4. Technique 3: Simplifying Complex Fractions

Sometimes an indeterminate form arises from a complex fraction (a fraction that has fractions in its numerator or denominator).

The strategy here is to first simplify the fraction into a standard rational form:

1. Find the least common denominator (LCD) of the "mini-fractions".
2. Combine the mini-fractions into a single fraction.
3. The expression should now be a standard rational function, where you can often use factorising and cancelling to find the limit.

The same video we've been using demonstrates this technique well.

Conclusion

Today we built a systematic strategy for solving limits that result in the indeterminate form $\frac{0}{0}$. Instead of giving up, we now have a powerful set of algebraic tools to resolve the ambiguity.

Key Takeaways:

• Always start with direct substitution. If it gives a real number, you are done.
• If you get $\frac{0}{0}$, it's an indeterminate form, which is a signal to do more work.
• Your algebraic toolkit includes:
  • Factorising and Cancelling: Your go-to for rational functions.
  • Rationalising with the Conjugate: Used for expressions involving square roots.
  • Simplifying Complex Fractions: Used to turn a fraction-within-a-fraction into a simpler form.
• After applying one of these techniques, the goal is always to create a new, equivalent expression where direct substitution now works.

In the next lesson, we will explore another type of limit: limits that involve infinity. We will investigate what happens to functions as $x$ grows infinitely large or small and introduce the concept of horizontal asymptotes, which you may have seen when studying rational functions.