Topic 1.7 Notes – Selecting Procedures for Determining Limits

How to Choose a Method to Evaluate a Limit

When you see $$\lim_{x \to a} f(x),$$ pause for a second. Look at what you’re given. The structure of the problem tells you the method.

There are four main scenarios in Unit 1:

1. A graph
2. A table
3. An expression where direct substitution works
4. An expression that gives $0/0$ and needs algebra

Let’s walk through each.

Limits from a Graph

If the problem gives you a graph, your job is visual.

In the example below, notice the hole at $x = 2$ around $y = 3$, and the filled-in dot at a different height. That difference is the whole point.

Limit vs. function value at x = 2

What you’re looking for:

• As $x \to a$, what y-value does the graph approach?
• Check both sides:
  • Left-hand limit
  • Right-hand limit

If both sides approach the same value, the limit exists.

Key situations:

• Hole (open circle) → limit can exist even if the function isn’t defined there.
• Jump → left and right approach different numbers → limit does not exist.
• Vertical asymptote → values grow toward $\infty$ or $-\infty$. That’s an infinite limit.

On multiple-choice questions, they love to trick you by putting the function value as an answer choice. The limit is about approach, not the filled-in dot. In the graph above, the limit is 3 even though $f(2) = 5$.

Limits from a Table

If you’re given numerical values, you’re estimating a trend.

You’ll see $x$-values getting closer to $a$ from:

• Slightly smaller numbers
• Slightly larger numbers

Then you watch the $y$-values.

What matters:

• Do both sides seem to approach the same number?
• Are the values growing very large positive or negative?
• Are left and right heading toward different numbers?

Don’t just grab the value at $x=a$. Sometimes it’s missing on purpose.

On calculator-active questions, you may generate your own table. Make sure you test values from both sides.

When Direct Substitution Works

If you plug in $x=a$ and get a normal real number, you’re done.

This works for:

• Polynomials
• Trig functions
• Exponential and logarithmic functions (in their domains)
• Rational functions where the denominator isn’t zero

Example idea:

$$\lim_{x\to 4} (3x^2 - 5x + 1)$$

Just substitute: $$3(4)^2 - 5(4) + 1 = 48 - 20 + 1 = 29.$$

That’s it.

Composite Functions

If you see $f(g(x))$:

1. Find $\lim g(x)$.
2. Plug that result into $f$.

This works as long as the outer function is continuous at that value.

When You Get $0/0$

If substitution gives $$\frac{0}{0},$$ that’s an indeterminate form. Now you simplify.

Common fixes:

1. Factor and Cancel

$$\frac{x^2 - 9}{x - 3}$$

Factor numerator: $$\frac{(x-3)(x+3)}{x-3}$$

Cancel the factor, then substitute.

Important: you can cancel factors, not individual terms.

2. Multiply by the Conjugate

Used when radicals are involved.

$$\frac{\sqrt{x+5} - 3}{x - 4}$$

Multiply top and bottom by $\sqrt{x+5} + 3$.

This removes the square root in the numerator so you can cancel.

3. Trig Identities

The big one you must know:

$$\lim_{x\to 0} \frac{\sin x}{x} = 1$$

If you can rewrite something to use this pattern, you’re set.

How to Think During a Test

Mentally run this checklist:

1. Is there a graph? → Read it.
2. Is there a table? → Look for trends.
3. Can I substitute directly?
   • If yes → stop.
4. Did I get $0/0$? → Simplify.
5. What structure do I see?
   • Factoring?
   • Conjugate?
   • Trig identity?

Efficiency matters. Some students try to factor everything automatically and waste time when substitution would have worked instantly.