Compute The Following Limits Calculator

This calculator helps you compute limits of functions as x approaches a specific value. Whether you're studying calculus or need to evaluate limits for practical applications, this tool provides a straightforward way to find the limit of a function.

How to Use This Calculator

To compute a limit using this calculator:

Enter the function you want to evaluate in the "Function" field. Use standard mathematical notation (e.g., "x^2 + 3x - 5").
Specify the value that x approaches in the "Approach from" field.
Select whether you want to evaluate a two-sided limit or a one-sided limit (left or right).
Click the "Calculate" button to compute the limit.

The calculator will display the result and provide an explanation of how the limit was determined.

Basic Limit Rules

Understanding these fundamental limit rules can help you evaluate limits more efficiently:

Sum/Difference Rule: lim(f(x) ± g(x)) = lim(f(x)) ± lim(g(x))
Product Rule: lim(f(x)g(x)) = lim(f(x)) * lim(g(x))
Quotient Rule: lim(f(x)/g(x)) = lim(f(x))/lim(g(x)) (if lim(g(x)) ≠ 0)
Constant Multiple Rule: lim(kf(x)) = k*lim(f(x))
Power Rule: lim(x^n) = (lim(x))^n

Note

These rules apply when the individual limits exist and are finite. Some functions may require more advanced techniques like L'Hôpital's Rule for evaluation.

One-Sided Limits

One-sided limits describe the behavior of a function as x approaches a value from either the left or the right:

Left-hand limit: lim(x→a⁻) f(x)
Right-hand limit: lim(x→a⁺) f(x)

If the left-hand and right-hand limits are equal, then the two-sided limit exists. If they are not equal, the two-sided limit does not exist.

Limit Definition

lim(x→a) f(x) = L means that for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Worked Examples

Example 1: Simple Polynomial

Compute lim(x→2) (x² + 3x - 5).

Using the limit rules, we substitute x = 2 directly into the function:

(2)² + 3(2) - 5 = 4 + 6 - 5 = 5

Therefore, lim(x→2) (x² + 3x - 5) = 5.

Example 2: Rational Function

Compute lim(x→3) (x² - 9)/(x - 3).

This is an indeterminate form (0/0), so we can factor the numerator:

(x² - 9) = (x - 3)(x + 3)

Thus, the limit becomes lim(x→3) (x - 3)(x + 3)/(x - 3) = lim(x→3) (x + 3) = 6.

Frequently Asked Questions

What is the difference between a limit and a function value?

A limit describes the behavior of a function as x approaches a certain value, while the function value is the actual output at that point. They can be different if the function is not defined at that point.

When does a limit not exist?

A limit does not exist if the left-hand and right-hand limits are not equal, if the function approaches infinity, or if the function oscillates as x approaches the point.

How do I evaluate limits at infinity?

For limits at infinity, you can often simplify the function by dividing numerator and denominator by the highest power of x in the denominator, then evaluate the limit as x approaches infinity.