Determining limits using algebraic properties of limits — AP Calculus BC
1. Basic Limit Laws and Direct Substitution ★☆☆☆☆ ⏱ 4 min
The fundamental algebraic properties of limits, called limit laws, let us break complex limits into simpler solvable parts. All laws assume that $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ both exist as finite real numbers.
- Constant multiple: $\lim_{x \to a} c f(x) = cL$ for any constant $c$
- Sum/difference: $\lim_{x \to a} [f(x) \pm g(x)] = L \pm M$
- Product: $\lim_{x \to a} [f(x) g(x)] = LM$
- Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, if and only if $M \neq 0$
- Power/root: $\lim_{x \to a} [f(x)]^n = L^n$ and $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L}$ (for even $n$, $L \geq 0$)
Exam tip: Always confirm the evaluation point is in the function's domain before using direct substitution—if it is, no extra work is needed.
2. Factoring and Canceling for 0/0 Indeterminate Forms ★★☆☆☆ ⏱ 3 min
When you substitute $x=a$ into a rational function and get $\frac{0}{0}$, you have an indeterminate form. This does not mean the limit does not exist—it only means the quotient law cannot be applied directly. $\frac{0}{0}$ almost always means the numerator and denominator share a common factor of $(x-a)$. Because we take the limit as $x \to a$, $x$ never actually equals $a$, so we can safely cancel the common factor and use direct substitution on the simplified expression.
Exam tip: If you get 0/0 after substitution, always look for a common linear factor first—9 times out of 10 on the AP exam, this factor cancels cleanly.
3. Rationalizing for Radical Indeterminate Forms ★★★☆☆ ⏱ 3 min
When 0/0 indeterminate forms include radicals, factoring will not work directly, so we use the method of rationalizing. This relies on the difference of squares identity: $(a - b)(a + b) = a^2 - b^2$. We multiply both numerator and denominator by the conjugate of the radical expression (the conjugate changes the sign between the radical term and the constant term, not inside the radical) to eliminate the radical, reveal a common factor, then cancel and substitute.
Exam tip: Always place the conjugate on the side that contains the radical—if the radical is in the numerator, multiply by the numerator's conjugate; if it is in the denominator, use the denominator's conjugate.
4. Algebraic Evaluation of Limits for Piecewise Functions ★★★☆☆ ⏱ 4 min
To find the limit as $x$ approaches a point where a piecewise function changes its rule, you evaluate the left-hand limit ($x \to a^-$) using the rule that applies for $x < a$, and the right-hand limit ($x \to a^+$) using the rule that applies for $x > a$. A two-sided limit exists if and only if both one-sided limits are equal. You use the same algebraic properties (direct substitution, factoring, rationalizing) to evaluate each one-sided limit separately.
Exam tip: Always double-check which piece corresponds to which side: $a^-$ means $x < a$, so use the rule for $x < a$, not the reverse.
Common Pitfalls
Why: Students confuse the function being undefined at $a$ with the limit not existing at $a$. 0/0 is an indeterminate form, not a final conclusion.
Why: Students confuse the limit as $x \to a$ with the value of the function at $a$.
Why: Students forget that 0/0 is indeterminate, while non-zero/zero has no finite limit.
Why: Students confuse the position of the sign change in the conjugate.
Why: Students mix up the notation for left and right limits.