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Evaluating Limits: Algebraic Manipulation

In the previous section, we introduced the first method to algebraically evaluate limits: direct substitution. We also discussed the limitations of this method, particularly when the function is not defined or discontinuous at the value we are evaluating the limit for. We will now explore different ways to manipulate the function algebraically to evaluate the limit.

Table of Contents​

Factoring​

Before we delve into the method of factoring, note this very important fact: factoring is not some magical technique that you should memorize for evaluating limits. In all of the "algebraic limit" techniques, it's just algebraic manipulation. Think of these more as some examples of how you can manipulate the function to make it easier to evaluate the limit.

With that said, let's consider the following limit:

If we try to evaluate this limit using direct substitution, we get , which is undefined. This function has a discontinuity at , and we need to simplify it before we can evaluate the limit. This is a clear indication that we need to simplify the function before we can evaluate the limit.

The numerator is a difference of squares, which can be factored as . Then, we can cancel out the common factor of in the numerator and denominator:

But wait! We can't just cancel out the . When we cancel something out, we are dividing both the numerator and the denominator by that factor. But at , this factor is , so we're dividing the numerator and denominator by . We will need to address this by adding a condition to our simplification. The following is the correct simplification (and a new function):

(This doesn't actually affect our calculations, but it's good to be aware of this.)

Here's the key point: even though is not defined at , the limit is. Hence, the limit is just evaluated at , which is .

Therefore, the limit is .

To summarize what we did:

  1. We want to evaluate the limit .
  2. We simplified the function by factoring the numerator for all except .
  3. This gave us a new function . We say that when .
  4. Because everywhere other than , we say that .

It's important to emphasize this again: and are not the same function. Given , we found a new function, , that is equal to everywhere except at .

Rationalizing​

Another common technique to simplify functions is rationalizing. This is particularly useful when you have a square root in the denominator. Let's consider the following limit:

If we try to evaluate this limit using direct substitution, we get , which is undefined.

To simplify this function, we can rationalize the numerator. To recap, rationalizing something involves multiplying by a clever form of to get rid of the square root. In this case, we can multiply by the conjugate of the numerator:

This simplifies to:

Now, we can cancel out the in the numerator and denominator, making sure to add a condition for :

Now, we can just evaluate the limit by substituting into the simplified function:

To summarize what we did:

  1. We want to evaluate the limit .
  2. We rationalized the numerator by multiplying by the conjugate of the numerator.
  3. This gave us a new function . We say that when .
  4. Because everywhere other than , we say that .

Trigonometric Limits​

Trigonometric limits are a common type of limit that can be simplified using trigonometric identities. Let's consider the following limit:

(Source)

If we try to evaluate this limit using direct substitution:

This is undefined, so we need to simplify the function. We can use the double-angle identity for cosine to simplify the denominator:

(There is a proof for this in the Appendix section.) By the Pythagorean identity, we know that . Hence:

We are now ready to simplify the original function:

Now, we can evaluate the limit by substituting into the simplified function:

Formalizing our Approach​

Here's a general approach to evaluating limits algebraically:

  1. We want to evaluate the limit .
  2. Through manipulation, we find a new function that is equal to everywhere except at .
  3. We say that when .
  4. Because everywhere other than , we say that .

The underlying mathematical principle here is that the limit of a function at a point is determined by the function's behavior around that point, not at that point itself. That's why the equality doesn't matter at the point ; it only matters in the neighborhood of . We can write this more formally as:

For a function and a limit , if there exists a function such that for all , then:

Summary and Next Steps​

In this section, we explored different algebraic techniques to evaluate limits.

Here are the key points to remember:

  • Factoring, rationalizing, and trigonometric identities are common techniques to simplify functions for limit evaluation.
  • The key idea is to find a new function that is equal to the original function everywhere except at the point where the limit is being evaluated.
  • The limit of a function at a point is determined by the function's behavior around that point, not at that point itself.

Appendix: Double-Angle Identity for Cosine​

Here's a quick geometric proof of the double-angle identity for cosine. Draw a circle with radius , and a horizontal line that splits the circle into two equal halves. Then, place a point somewhere on the circle and draw a line from the both ends of the horizontal line to the point.

First, consider the triangle . Its hypotenuse is the horizontal line, . By SOHCAHTOA, we have and .

Next, consider the triangle . Its hypotenuse is the line from the left to the point, . By SOHCAHTOA again, we see that . Hence, .

Finally, consider the triangle . The length is equal to . Apply SOHCAHTOA again on this triangle to get . is just the radius of the circle, which is . Hence:

This is the double-angle identity for cosine.

Appendix: Proof of our General Approach​

Let's prove the general approach we outlined earlier. We want to show that if for all , then . Before proceeding, note:

  1. One should not need this proof to convince oneself of this fact. It should make complete sense intuitively.
  2. This proof is more for the sake of completeness and rigor.

Let . This means that for every , there exists a such that if , then . The proof is simply to show that the same works for as well.

Since for all , we know that for all . Then:

This is true for all , so the same that works for also works for . Hence, .