Limit Calculator
Calculate limits of functions as x approaches any value
Use: x^2, sqrt(x), sin(x), cos(x), tan(x), ln(x), e^x, pi
The value that x is approaching
lim(x→) f(x) = lim(x→⁻) f(x) = lim(x→⁺) f(x) =
Does Not Exist
Limit exists Left and right limits differ
Left-Hand Limit (x→a⁻)
Right-Hand Limit (x→a⁺)
Approach Table
From Left (x < a)
| x | f(x) |
|---|---|
From Right (x > a)
| x | f(x) |
|---|---|
Trigonometric Limits
lim(x→0) sin(x)/x = 1
lim(x→0) (1-cos(x))/x = 0
lim(x→0) tan(x)/x = 1
Exponential & Log
lim(x→0) (eˣ-1)/x = 1
lim(x→0) ln(1+x)/x = 1
lim(x→∞) (1+1/x)ˣ = e
Polynomial Limits
lim(x→a) xⁿ = aⁿ
lim(x→∞) 1/x = 0
lim(x→0⁺) 1/x = ∞
Indeterminate Forms
0/0, ∞/∞, 0·∞
∞-∞, 0⁰, ∞⁰, 1^∞
Use L'Hôpital's Rule
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About Limit Calculator
What is a Limit?
A limit describes the value that a function approaches as the input (x) approaches a specific value. Limits are fundamental to calculus and are used to define derivatives, integrals, and continuity. The notation lim(x→a) f(x) = L means that f(x) gets arbitrarily close to L as x approaches a.
How to Use This Calculator
- Enter Function: Type your function f(x) using standard mathematical notation
- Enter X Approaches: Specify the value that x is approaching
- Select Direction: Choose left-sided, right-sided, or two-sided limit
- Calculate: Click calculate to see the limit value and approach table
Understanding Your Results
Two-Sided Limit
The limit exists only if both left-hand and right-hand limits exist and are equal. This is notated as lim(x→a) f(x).
One-Sided Limits
- Left-hand limit lim(x→a⁻): The value approached as x comes from values less than a
- Right-hand limit lim(x→a⁺): The value approached as x comes from values greater than a
Indeterminate Forms
Some limits produce indeterminate forms like 0/0 or ∞/∞. These require special techniques like L'Hôpital's Rule or algebraic manipulation to evaluate.
Common Limit Formulas
| Limit | Value |
|---|---|
| lim(x→0) sin(x)/x | 1 |
| lim(x→0) (1-cos(x))/x | 0 |
| lim(x→0) (eˣ-1)/x | 1 |
| lim(x→∞) (1+1/x)ˣ | e |
| lim(x→0) tan(x)/x | 1 |
| lim(x→0) ln(1+x)/x | 1 |
Frequently Asked Questions (FAQ)
What is the difference between one-sided and two-sided limits?
A two-sided limit requires the function to approach the same value from both directions. One-sided limits only consider approach from one direction.
When does a limit not exist?
A limit doesn't exist when: (1) left and right limits are different, (2) the function oscillates infinitely, or (3) the function approaches infinity.
How accurate is this calculator?
The numerical method evaluates the function at points very close to the target value, providing precision to approximately 6-8 decimal places for smooth functions.
Important Limitations
- Cannot evaluate symbolic limits requiring algebraic simplification
- May have numerical issues at points where the function is undefined
- Works best with continuous functions near the point of interest
Disclaimer: This calculator is for educational purposes only. For complex limit evaluations, verify results with symbolic mathematics software.