Direct Substitution Method

The Direct Substitution Method

The easiest way to find a limit? Just plug in the number!

When It Works

Direct substitution works when the function is continuous at the point. This includes:

• Polynomial functions: $x^2$, $3x^3 - 2x + 1$, etc.
• Rational functions (when denominator ≠ 0)
• Radical functions (when defined)
• Trigonometric functions (at most points)
• Exponential and logarithmic functions

If you can evaluate $f(a)$ without any problems, then $\lim_{x \to a} f(x) = f(a)$

The Process

To find $\lim_{x \to a} f(x)$:

1. Try substituting x = a directly into f(x)
2. If you get a number → That's your answer! ✓
3. If you get $\frac{0}{0}$, $\frac{\infty}{\infty}$, etc. → Need a different technique

Example 1: Polynomial

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

Solution: Just substitute x = 2:

$\lim_{x \to 2} (x^2 + 3x - 1) = (2)^2 + 3(2) - 1 = 4 + 6 - 1 = 9$

✓ Answer: 9

Example 2: Rational Function (Works)

Find $\lim_{x \to 3} \frac{x + 1}{x - 1}$

Solution: Substitute x = 3:

$\lim_{x \to 3} \frac{x + 1}{x - 1} = \frac{3 + 1}{3 - 1} = \frac{4}{2} = 2$

✓ Answer: 2

Example 3: When It Fails

Find $\lim_{x \to 1} \frac{x - 1}{x^2 - 1}$

Attempt: $\frac{1 - 1}{1^2 - 1} = \frac{0}{0}$

✗ This is indeterminate! Direct substitution doesn't work here.

When you get $\frac{0}{0}$, you need algebraic manipulation (factoring, rationalizing, etc.)

Indeterminate Forms

These special cases mean direct substitution has failed:

We'll cover techniques for these in upcoming lessons!

Quick Check Method

Ask yourself: "Can I safely plug in this number?"

• Is the function defined there? → Try it!
• Does the denominator become zero? → Can't use direct substitution
• Is there a square root of a negative? → Can't use direct substitution
• Otherwise → Go for it!

Practice Tip

Always try direct substitution first. It's the fastest method when it works!

If you get a real number, you're done. If you get an indeterminate form, move on to other techniques.