Evaluating Limits

The limit laws say you can usually find a limit by substituting. This lesson is about the times you can’t, when substitution gives . That’s not a dead end. It’s a sign that the top and bottom share a factor that’s zero at , and evaluating limits mostly means finding that factor and cancelling it. Once you know how to find a limit this way, you’ll recognise the same moves inside every derivative you work out.

How to Evaluate a Limit: the Strategy

1. Substitute first. If you get a number, you’re finished.
2. If you get a nonzero number over 0, the function grows without bound and the limit does not exist (check one-sided limits to describe how).
3. If you get , rewrite the expression using the technique that fits, cancel, and substitute again.

Why is cancelling allowed? A limit never uses the value at , only values near it. For every , the factor isn’t zero, so dividing it out changes nothing the limit can see.

Technique 1: Factor and Cancel

Find .

Substituting gives . Since makes the top zero, must be a factor of it:

The same works for a difference of cubes. With :

If the factoring is rusty, the factoring polynomials lesson covers every pattern you’ll need.

Technique 2: Rationalize With the Conjugate

Find .

Substitution gives , and there’s nothing to factor. Instead, multiply the top and bottom by the conjugate , just as you would when rationalizing denominators:

Now substitute: .

Technique 3: Combine the Fractions

Find .

When small fractions sit inside a big one, put them over a common denominator first:

Dividing by cancels the on top, leaving . Substitute: .

Technique 4: Expand and Cancel

Find .

Expand the bracket: . Cancel and substitute to get 4. Limits in exactly this form are how derivatives are defined, and they’re the same calculation as the difference quotient from algebra, with at the end.

Special Trig Limits

Trig expressions often don’t factor, so two known limits do the work (with in radians):

To use them, make the expression match. For , multiply top and bottom by 3:

because too.

Tangent works the same way once you write it with sine and cosine. Since , and the two pieces approach 1 and 1, .

The conjugate idea even works for trig. To find , multiply top and bottom by . The top becomes , so the expression is , which approaches .

The first of these special limits is proved with the squeeze theorem in the next lesson.

A Limit That Needs Two Techniques

Some limits need more than one move. Find .

Substitution gives . The square root on the bottom calls for the conjugate, so multiply top and bottom by . The bottom becomes :

Now the polynomial on top factors as , and the cancels, leaving . Substitute: . The order doesn’t matter much; what matters is to keep rewriting until the factor that makes both top and bottom zero disappears.

Checking Your Answer

Algebra is easy to slip on, so a quick numerical check is worth the ten seconds it takes. Plug in a value very close to on each side. For the conjugate example, gives about 0.16662 and gives about 0.16671. Both sit right next to , so the answer is almost certainly right. For the two-technique example, gives about 4.003 and about 3.997, closing in on 4. If your check disagrees with your algebra, look for a sign error or a factor you cancelled that wasn’t really a factor.

When Nothing Cancels

If substitution gives instead of , no algebra will rescue it. In , the top heads for 5 and the bottom for 0, so the function blows up and the limit does not exist. Expressions with absolute values, such as , need separate one-sided limits, and later in this chapter L’Hôpital’s rule gives one more tool for stubborn forms.

Common Mistakes to Avoid

• Stopping at 0/0. It’s never the answer; it’s the cue to simplify.
• Cancelling terms instead of factors. In you can’t cancel the ‘s or the 2’s; factor first.
• Expanding the conjugate on the bottom. Leave factored so the cancels.
• Using degrees in trig limits. is only true in radians.

Practice Problems

Work each one before opening the answer.

Problem 1. Find .

Problem 2. Find .

Problem 3. Find .

Problem 4. Find .

Problem 5. Find .

Next, the squeeze theorem: how to pin down a limit by trapping a function between two others, and the proof that .