Evaluate The Following Limit Calculator

Limits are fundamental in calculus for understanding the behavior of functions as they approach specific points. This calculator helps you evaluate limits of functions as x approaches a given value, whether finite or infinite.

How to Use This Calculator

To evaluate a limit using our calculator:

Enter the function you want to evaluate in the "Function" field. Use standard mathematical notation (e.g., x^2, sin(x), e^x).
Specify the point x approaches in the "Approach from" field.
Select whether x approaches from the left, right, or both sides.
Click "Calculate" to see the result.

The calculator will display the limit value if it exists, or indicate if the limit does not exist.

Types of Limits

There are several types of limits you may encounter:

Finite limits: The function approaches a finite value as x approaches a point.
Infinite limits: The function grows without bound as x approaches a point.
One-sided limits: The function approaches different values from the left and right sides.
Indeterminate forms: Limits that result in forms like 0/0 or ∞/∞, which may require further analysis.

Limit Laws

These fundamental laws help simplify limit calculations:

Sum/Difference Law

lim (f(x) ± g(x)) = lim f(x) ± lim g(x)

Product Law

lim (f(x) · g(x)) = lim f(x) · lim g(x)

Quotient Law

lim (f(x)/g(x)) = lim f(x)/lim g(x) if lim g(x) ≠ 0

Constant Multiple Law

lim (c·f(x)) = c·lim f(x)

Common Limit Examples

Here are some standard limit examples:

Function	Limit as x→a	Result
x	a	a
sin(x)	0	0
e^x	0	1
1/x	∞	0

Frequently Asked Questions

What if the limit doesn't exist?

If the left and right limits are not equal, or if the function approaches infinity, the limit does not exist. The calculator will indicate this case.

How do I handle limits at infinity?

For limits at infinity, enter "Infinity" in the "Approach from" field and select the appropriate direction.

What if I get an indeterminate form?

Indeterminate forms like 0/0 or ∞/∞ require techniques like L'Hôpital's Rule or algebraic manipulation to evaluate.