Vertical Asymptote Limits: Step-by-Step Guide
A vertical asymptote at x = a occurs when f(x) tends to plus or minus infinity as x approaches a. This typically happens when the denominator of a rational function is zero while the numerator is nonzero. The sign of the infinity is determined by examining the signs of numerator and denominator on each side of a.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Substitute and check the form. Plug x = a into f(x). If the result is (nonzero)/0, x = a is a vertical asymptote and the limit is plus or minus infinity. If the result is 0/0, do not conclude an asymptote — try factoring instead.\[f(a) = \frac{c}{0}, \;\; c \neq 0\]
- Examine signs from each side. Pick a value just less than a and just greater than a. Determine the sign of the numerator and denominator at each. The sign of the quotient tells you whether the limit is +infinity or -infinity from that side.\[\text{sign}\left(\frac{\text{num}(a^\pm)}{\text{den}(a^\pm)}\right)\]
- State the one-sided limits. Write lim x->a- f(x) and lim x->a+ f(x) separately as plus or minus infinity based on the sign analysis.\[\lim_{x \to a^-} f(x), \;\; \lim_{x \to a^+} f(x)\]
- Decide the two-sided limit. If both one-sided limits agree (both +infinity or both -infinity), the two-sided limit equals that value. Otherwise the two-sided limit does not exist (DNE).\[\lim_{x \to a} f(x) = \begin{cases} \pm\infty & \text{if both sides agree} \\ \text{DNE} & \text{otherwise} \end{cases}\]
Worked example
Evaluate lim x->2 1/(x - 2)^2 and lim x->2 1/(x - 2).
Step 1: For 1/(x-2)^2, substitution gives 1/0; the denominator (x-2)^2 is always nonnegative.
Step 2: As x -> 2 from either side, (x-2)^2 -> 0+, so 1/(x-2)^2 -> +infinity. Both sides agree, so lim x->2 1/(x-2)^2 = +infinity.
Step 3: For 1/(x-2), substitution also gives 1/0, but (x-2) changes sign across 2.
Step 4: As x -> 2- (x slightly less than 2), x - 2 < 0, so 1/(x-2) -> -infinity. As x -> 2+ (x slightly greater than 2), x - 2 > 0, so 1/(x-2) -> +infinity.
Step 5: One-sided limits disagree, so lim x->2 1/(x-2) does not exist (DNE).
Verification: 1/(1.99 - 2) = -100 and 1/(2.01 - 2) = 100, confirming opposite infinities; whereas 1/(1.99-2)^2 = 1/(2.01-2)^2 = 10000, both large positive.
Tips and common mistakes
- Vertical asymptotes appear at zeros of the denominator that are NOT also zeros of the numerator.
- If both numerator and denominator are zero at a, the discontinuity may be removable — factor first.
- Always test a value slightly less than a and slightly greater than a to determine signs.
- Even powers of (x - a) keep the denominator positive on both sides; odd powers flip sign.
- If left and right limits both equal +infinity (or both -infinity), the two-sided limit is that infinity.
- If one side is +infinity and the other is -infinity, the two-sided limit does not exist.
- Saying a limit equals infinity means it does not exist as a finite number, but it is more informative than merely DNE.
- Vertical asymptotes are graph features — drawing a quick sign chart is often faster than algebra.