Why Limits Trip Up Students
Limits are conceptually subtle — they describe behaviour near a point rather than at it. This distinction causes systematic errors that appear over and over in student work. The good news: once you understand the root cause of each mistake, you are unlikely to make it again.
01
Treating the Limit as the Function Value
❌ Wrong
lim (x→2) [(x²−4)/(x−2)] — "f(2) is undefined, so the limit is undefined."
✓ Correct
Factor: (x−2)(x+2)/(x−2) = x+2 for x ≠ 2. lim = 2+2 = 4.
The limit asks what f(x) approaches as x → 2, not what f(2) equals. The function being undefined at x = 2 is irrelevant to the limit — that is exactly the situation limits were invented for.
02
Illegal Cancellation in Limits
❌ Wrong
lim (x→0) [(x² + x) / x] — "Cancel the x² with the x: lim = x+1 → 1+1 = 2."
✓ Correct
Factor: x(x+1)/x = x+1 for x ≠ 0. lim (x→0) [x+1] = 0+1 = 1.
You cannot cancel x² with x directly. Factor out x from the numerator first: x(x+1)/x. Now cancel the common factor x. The result x+1 has limit 1 as x→0, not 2.
03
Applying L'Hôpital's Rule Without Checking the Form
❌ Wrong
lim (x→2) [(x²+1)/(x+3)] — applies L'Hôpital: [2x/1] = 4. But f(2) = 5/5 = 1!
✓ Correct
Direct substitution: (4+1)/(2+3) = 5/5 = 1. L'Hôpital only applies to 0/0 or ∞/∞.
L'Hôpital's Rule is only valid for 0/0 and ∞/∞ indeterminate forms. Applying it to a limit that can be directly evaluated gives the wrong answer. Always check the form first.
04
Ignoring One-Sided Limits at Jump Discontinuities
❌ Wrong
For f(x) = |x|/x at x=0: "direct sub gives 0/0, so limit = 0."
✓ Correct
lim (x→0⁺) = +1. lim (x→0⁻) = −1. One-sided limits differ → two-sided limit DNE.
When you suspect a piecewise or absolute-value function, always check both one-sided limits separately. If they differ, the two-sided limit does not exist — writing "0" or any number is wrong.
05
Assuming ∞ − ∞ = 0
❌ Wrong
lim (x→∞) [x² − x] = ∞ − ∞ = 0
✓ Correct
Factor: x(x−1). As x→∞, both factors → ∞. Limit = ∞, not 0.
∞ − ∞ is an indeterminate form — it can equal anything, including 0, ∞, or any finite number. You must factor or algebraically manipulate the expression before concluding. Never "cancel" infinities arithmetically.
06
Misapplying the sin(x)/x Trig Limit
❌ Wrong
lim (x→0) [sin(3x) / (5x)] = 1, because "sin(x)/x = 1."
✓ Correct
Rewrite: (3/5)·[sin(3x)/(3x)] → (3/5)·1 = 3/5.
The rule lim (x→0) [sin(x)/x] = 1 only works when the argument of sin matches the denominator exactly. When they differ, you must rewrite by multiplying and dividing by the appropriate constant first.
07
Declaring DNE Too Quickly for Infinite Limits
❌ Wrong
lim (x→0) [1/x²] = DNE, because "you cannot divide by 0."
✓ Correct
From both sides: 1/x² → +∞. So lim = +∞ (both sides agree). Write lim = +∞.
A limit of +∞ is technically "does not exist" as a real number, but it carries precise information. When both one-sided limits agree and go to +∞ (or both to −∞), you should state the limit is +∞ or −∞, not just "DNE" — the latter is imprecise and loses marks.
08
Misapplying the Degree Rule for Limits at Infinity
❌ Wrong
lim (x→∞) [(3x³ + x) / (2x²)] = 3/2, because "same degree, take leading coefficients."
✓ Correct
deg(num) = 3 > deg(den) = 2 → limit = ∞, not 3/2.
The leading-coefficient rule only applies when numerator and denominator have equal degree. If the numerator has higher degree, the limit is ±∞. If lower, the limit is 0. Always check degrees before applying the shortcut.
Before-You-Submit Checklist
✓Did I check whether direct substitution works first?
✓If factoring, did I factor correctly and verify the cancellation?
✓If using L'Hôpital's, did I verify the form is 0/0 or ∞/∞?
✓For piecewise/absolute-value functions, did I check both sides?
✓For trig limits, does sin match the denominator exactly?
✓For limits at infinity, did I check the degree correctly?
✓Did I state the limit is ±∞ precisely (not just "DNE") where appropriate?
Frequently Asked Questions
How do I know which technique to use for a limit?
Follow the decision flowchart in the How to Solve Limits article: (1) Try direct substitution. (2) If 0/0 and polynomial, try factoring. (3) If square roots, try rationalising. (4) If trig near 0, check standard forms. (5) If still stuck with 0/0 or ∞/∞, use L'Hôpital's Rule.
What is the difference between DNE and ∞?
Saying a limit is ∞ is more informative than DNE — it tells you exactly how the function is behaving (growing without bound). DNE is the correct technical answer (∞ is not a real number), but in calculus courses you should specify ±∞ when that is the behaviour, and reserve "DNE" for cases where the limit is undefined for a different reason (like oscillation or a two-sided conflict).
Should I always verify my limit answer?
Yes — especially on exams. The best verification is to substitute values of x very close to a (from both sides) into the original function and check that the outputs approach your claimed limit. This numerical check catches most errors immediately and takes only seconds.
References & Further Reading
- Stewart, J. (2015). Calculus, §2.2–2.6. Cengage.
- College Board (2024). AP Calculus AB and BC Course and Exam Description.
- Larson, R. & Edwards, B. (2013). Calculus, 10th ed. Cengage.