This works for:

• Polynomials
• Rational functions (when denominator ≠ 0)
• Radicals (when defined)
• Trig functions (at most points)

Example: Direct Substitution

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

Just substitute $x = 2$: $= 3(2)^2 - 5(2) + 1 = 12 - 10 + 1 = 3$

When Direct Substitution Fails

If you get:

• $\frac{0}{0}$ → Indeterminate form (use algebra!)
• $\frac{c}{0}$ where $c \neq 0$ → Infinite limit or DNE
• $\frac{c}{d}$ where $d \neq 0$ → That's your answer!

Properties of Limits

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$:

Sum/Difference: $\lim_{x \to a} [f(x) \pm g(x)] = L \pm M$

Product: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$

Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$ (if $M \neq 0$)

Constant Multiple: $\lim_{x \to a} [c \cdot f(x)] = c \cdot L$

Power: $\lim_{x \to a} [f(x)]^n = L^n$

Strategy

1. Try direct substitution first
2. If you get a real number → Done!
3. If you get $\frac{0}{0}$ → Use algebraic techniques (next topic!)
4. If you get $\frac{c}{0}$ → Investigate the limit more carefully