Free Limit Calculator with Steps & Graph
Evaluate limits step-by-step with an interactive graph and convergence table. Direct substitution, L'Hôpital's rule, factoring, and special limits. Perfect for A-Level and university calculus.
Step-by-step learning with explanations
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Enter a mathematical expression above to learn how to evaluate the limit step-by-step.
Visualizations appear here after evaluation
Animated convergence, interactive graph, and value table
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What is a Limit?
A limit describes the value a function approaches as the input gets closer and closer to a particular value. It is written as lim(x→a) f(x) = L, meaning "as x gets arbitrarily close to a, f(x) gets arbitrarily close to L".
Formal Definition (ε-δ)
For every epsilon (however small), there exists a delta such that f(x) is within epsilon of L whenever x is within delta of a.
Limits are the foundation of calculus. They define derivatives (rate of change as Δx → 0) and integrals (area as strip width → 0), making them the most fundamental concept in all of calculus.
Intuitive analogy: Imagine walking toward a door.
You take half the remaining distance each step.
You never reach the door, but the limit of your position is the door.
Limit Notation
Limit Laws & Properties
If lim f(x) = L and lim g(x) = M as x → a, then these rules apply:
Sum Rule
lim [f(x) + g(x)] = L + M
Difference Rule
lim [f(x) − g(x)] = L − M
Product Rule
lim [f(x) · g(x)] = L · M
Quotient Rule
lim [f(x)/g(x)] = L/M (if M ≠ 0)
Constant Multiple
lim [c · f(x)] = c · L
Power Rule
lim [f(x)]ⁿ = Lⁿ
Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) and lim g = lim h = L, then lim f = L
Types of Limits
| Type | Notation | Example |
|---|---|---|
| Two-sided | lim(x→a) f(x) | lim(x→2) x² = 4 |
| Left-hand | lim(x→a⁻) f(x) | lim(x→0⁻) 1/x = −∞ |
| Right-hand | lim(x→a⁺) f(x) | lim(x→0⁺) 1/x = +∞ |
| At infinity | lim(x→∞) f(x) | lim(x→∞) 1/x = 0 |
| Infinite limit | lim(x→a) f(x) = ±∞ | lim(x→0) 1/x² = +∞ |
Indeterminate Forms Deep Dive
These seven forms arise when direct substitution fails. Each requires a specific resolution technique.
Factor, L'Hôpital, rationalize, or special limits
L'Hôpital's rule or divide by dominant term
Rewrite as 0/0 or ∞/∞ using reciprocals
Combine into a single fraction, then simplify
Take ln, evaluate lim of exponent × ln(base)
Use lim = e^(lim (f-1)·g) pattern
Take ln, converts to 0 · ∞ form
Method Decision Guide
Follow this decision tree to choose the right technique:
Try Direct Substitution
Plug in x = a. If you get a finite number, you're done!
Identify the Form
If you get 0/0, ∞/∞, or another indeterminate form, note which one.
For 0/0: Try Factoring First
Factor numerator and denominator, cancel common (x − a) terms, then substitute again.
For 0/0 or ∞/∞: Apply L'Hôpital's Rule
Differentiate numerator and denominator separately. Re-evaluate. Repeat if needed.
Check Special Limits
sin(x)/x → 1, (eˣ−1)/x → 1, (1−cos(x))/x → 0 as x → 0. Memorise these!
For Infinity: Compare Degrees
For rational functions at infinity, compare polynomial degrees of top and bottom.
Common Mistakes in Limit Problems
Avoid these frequent errors in A-Level and university exams:
Applying L'Hôpital's Rule when it's not 0/0 or ∞/∞
L'Hôpital only works for 0/0 and ∞/∞. Using it for other forms (like 0/1 or 1/0) gives wrong answers.
Always verify the indeterminate form BEFORE differentiating.
Differentiating as a quotient rule instead of separately
L'Hôpital says: differentiate numerator alone AND denominator alone. It is NOT the quotient rule d/dx[f/g].
f'(x)/g'(x), not [f'g - fg']/g².
Forgetting to check one-sided limits for DNE
A two-sided limit only exists if both one-sided limits exist AND are equal. Students often forget to check both.
When in doubt, evaluate the left-hand and right-hand limits separately.
Assuming continuous = has limit (and vice versa)
A function can have a limit at a point where it's not defined (removable discontinuity). And a defined function can lack a limit (jump discontinuity).
The limit depends on APPROACH behavior, not the function value at the point.
Wrong sign for one-sided infinity
Students often write +∞ when the answer is −∞. The sign depends on which side you approach from.
Test a value slightly to the left/right of the approach point to determine the sign.
Not simplifying before substituting
Jumping to L'Hôpital when simple factoring would cancel the 0/0 in one step.
Always try factoring and canceling first — it's faster and less error-prone.
Worked Examples
Practice with these GCSE, A-Level, and university-level limit problems:
Example 1: lim(x→3) x² + 2x − 1
Try direct substitution: f(3) = 3² + 2(3) − 1
= 9 + 6 − 1
= 14
Direct substitution works because the function is a polynomial (continuous everywhere).
Example 2: lim(x→1) (x² − 1)/(x − 1)
Direct sub: f(1) = (1−1)/(1−1) = 0/0 (indeterminate!)
Factor: (x²−1) = (x+1)(x−1)
Cancel: (x+1)(x−1)/(x−1) = x+1
Substitute: 1 + 1
= 2
Example 3: lim(x→0) sin(x)/x
Direct sub: sin(0)/0 = 0/0 (indeterminate!)
L'Hôpital: d/dx[sin(x)] / d/dx[x] = cos(x)/1
Substitute: cos(0) = 1
= 1
This is also a special limit worth memorising: lim(x→0) sin(x)/x = 1.
Example 4: lim(x→∞) (3x² + 1)/(2x² − 5)
Both numerator and denominator → ∞ (form ∞/∞)
Compare degrees: both are degree 2
Ratio of leading coefficients: 3/2
= 3/2 = 1.5
For equal-degree rational functions, the limit at infinity is always the ratio of leading coefficients.
Special Limits Reference Table
These must-know limits appear frequently in exams and proofs. Memorise them!
| Limit | Value | Name |
|---|---|---|
| lim(x→0) sin(x)/x | 1 | Sine limit |
| lim(x→0) (1−cos(x))/x | 0 | Cosine limit |
| lim(x→0) (1−cos(x))/x² | 1/2 | Cosine squared limit |
| lim(x→0) tan(x)/x | 1 | Tangent limit |
| lim(x→0) (eˣ−1)/x | 1 | Exponential limit |
| lim(x→0) ln(1+x)/x | 1 | Logarithm limit |
| lim(x→∞) (1+1/x)ˣ | e ≈ 2.718 | Definition of e |
| lim(x→0) (1+x)^(1/x) | e ≈ 2.718 | Alternative def of e |
How to Evaluate Limits
Direct Substitution
Try plugging in the value directly. If you get a real number (not 0/0 or ∞/∞), that's your answer! This works for all polynomials and most elementary functions at points where they're continuous.
Factoring & Simplifying
For 0/0 forms, factor the numerator and denominator. Look for common factors like (x − a) to cancel. Then try direct substitution again on the simplified expression.
L'Hôpital's Rule
For 0/0 or ∞/∞ forms only: differentiate the numerator and denominator SEPARATELY (not quotient rule!), then re-evaluate. Repeat if you get another indeterminate form.
Special Limits
Memorise key limits: sin(x)/x → 1, (1−cos(x))/x → 0, (eˣ−1)/x → 1, (1+1/x)ˣ → e. These appear frequently and save time in exams.
Squeeze Theorem
When you can't evaluate directly, bound f(x) between two simpler functions that have the same limit. If g(x) ≤ f(x) ≤ h(x) and lim g = lim h = L, then lim f = L.
Frequently Asked Questions
How do I evaluate a limit?
Start with direct substitution. If indeterminate, try factoring, L'Hôpital's rule, or algebraic manipulation. Our calculator automatically selects the best method.
What does "limit does not exist" mean?
The function approaches different values from left and right, oscillates without settling, or diverges in different directions from each side.
When do I use L'Hôpital's rule?
Only when direct substitution gives 0/0 or ∞/∞. Differentiate the numerator and denominator separately, then re-evaluate.
What is lim(x→0) sin(x)/x?
This famous limit equals 1. It's fundamental to calculus and used in proving derivative formulas for trigonometric functions.
What are one-sided limits?
Left limit (x→a⁻) approaches from below; right limit (x→a⁺) approaches from above. Both must match for the two-sided limit to exist.
What are the 7 indeterminate forms?
0/0, ∞/∞, 0·∞, ∞−∞, 0⁰, 1^∞, and ∞⁰. Each requires specific techniques to resolve.
How does the graph help?
The interactive graph shows function behavior near the limit point, with dashed lines at x = a and y = L. You can see discontinuities, asymptotes, and convergence visually.
Is this suitable for A-Level revision?
Yes! Covers all A-Level limit techniques with step-by-step Learn Mode, graph visualization, and exam-style practice questions.
What is the Squeeze Theorem?
If g(x) ≤ f(x) ≤ h(x) near x = a and lim g(x) = lim h(x) = L, then lim f(x) = L. Useful when direct evaluation is difficult.
Can I use this for university calculus?
Absolutely. The calculator handles all standard limit techniques including L'Hôpital, infinity analysis, and indeterminate forms needed for university-level analysis courses.
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