Removable discontinuity
Limit of (x^2-1)/(x-1) as x approaches 1
Method: Factor before substituting.
- 1Direct substitution gives 0/0.
- 2Factor x^2-1 as (x-1)(x+1).
- 3Cancel for nearby x and evaluate x+1 at 1.
Final answer
Free online calculator
Limit Calculator
Evaluate one-sided and two-sided limits with detailed reasoning about behavior near the target point.
Limit mode
Limit
=
Step-by-step solutions
Clear, detailed explanations
Instant results
Accurate answers in seconds
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How to Use the Limit Calculator
1
Enter the limit
Type the function and the value x is approaching.
2
Analyze behavior
We examine the function from both sides of the point.
3
Get the result
Receive the limit value or a clear statement that it does not exist.
Example
How the Limit Calculator Chooses a Method
Evaluate nearby behavior at a point or at infinity and understand how an indeterminate form is resolved.
The calculator tries direct substitution first. If that produces an indeterminate form, it simplifies algebraically or applies a suitable standard limit, while one-sided behavior is checked near discontinuities.
Supported problem types
- Finite and infinite limits
- Rational, trigonometric, exponential, and logarithmic expressions
- Removable discontinuities and one-sided behavior
Limits and assumptions
- A two-sided limit does not exist when the left and right limits disagree.
- Oscillatory expressions may remain without a single limiting value.
Worked Limit Calculator Examples
Each example names the method, shows the ordered reasoning, and keeps the final answer separate so you can check your own work.
Standard trigonometric limit
Limit of sin(x)/x as x approaches 0
Method: Use the fundamental sine limit.
- 1Recognize the 0/0 form.
- 2Apply the standard limit in radians.
- 3Conclude the nearby ratio approaches 1.
Final answer
Behavior at infinity
Limit of (3x^2+1)/(x^2-4) as x approaches infinity
Method: Compare the highest powers.
- 1Divide numerator and denominator by x^2.
- 2Terms containing 1/x^2 approach zero.
- 3Take the ratio of leading coefficients.
Final answer
Learn This Topic
What the Limit Calculator Helps You Understand
Evaluate one-sided and two-sided limits with detailed reasoning about behavior near the target point.
Learning focus 1
Type the function and the value x is approaching.
Learning focus 2
We examine the function from both sides of the point.
Learning focus 3
Receive the limit value or a clear statement that it does not exist.
Input tips
- • Type the function and the value x is approaching.
- • Yes. Specify left or right in your input and the solver will compute the one-sided limit.
Common mistakes to avoid
- • Entering ambiguous notation without parentheses.
- • Using a final result without checking that it matches the original problem.
Frequently Asked Questions
Can I compute one-sided limits?
Yes. Specify left or right in your input and the solver will compute the one-sided limit.
What if the limit does not exist?
The solver will state that the limit does not exist and explain why, such as divergence or unequal one-sided limits.
Does 0/0 mean the limit is zero?
No. It is an indeterminate form that signals the expression needs simplification or another limiting method.
When does a two-sided limit fail to exist?
It fails when either one-sided limit is missing or when the left and right limiting values are different.
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