Example 1, Direct substitution
lim x→2 (x² + 3)
7
- 1
Substitute x = 2
(2)² + 3 = 4 + 3 - 2
Result
7
If the function is continuous at the point, just plug in.
Finite · Infinite · One-sided
Limit Calculator lim x→a f(x)
Evaluate limits at finite points and at infinity, including indeterminate forms via L'Hôpital's rule. Step-by-step solutions for five common limit types.
For complex problems, use the photo solver.
What a limit means
L'Hôpital's Rule
lim f/g = lim f'/g' (when 0/0 or ∞/∞)
When direct substitution gives an indeterminate form, differentiate the top and bottom separately, then try the limit again. Apply repeatedly if needed.
A limit asks: what value does f(x) approach as x gets arbitrarily close to some target? It might be a finite number, infinity, or might not exist. Limits are the foundation that makes derivatives and integrals rigorous.
When direct substitution gives an indeterminate form (0/0, ∞/∞, 0·∞, ∞ − ∞, 1^∞, 0⁰, ∞⁰), use algebraic manipulation (factor, rationalize) or L'Hôpital's rule (differentiate top and bottom separately) to resolve it.
Worked Examples
Five limits across every common type
Direct substitution, factor-and-cancel, infinity, L'Hôpital's rule, and one-sided limits.
Example 2, Factor and cancel
lim x→3 (x² − 9)/(x − 3)
6
- 1
Factor numerator
x² − 9 = (x − 3)(x + 3) - 2
Cancel (x − 3)
lim x→3 (x + 3) - 3
Substitute
= 3 + 3 = 6
Direct substitution gives 0/0 (indeterminate); factor to simplify.
Example 3, Limit at infinity
lim x→∞ (3x² + 1)/(2x² + 5x)
3/2
- 1
Divide by x²
(3 + 1/x²) / (2 + 5/x) - 2
As x → ∞
1/x² → 0, 5/x → 0 - 3
Result
3 / 2
Divide top and bottom by the highest power of x.
Example 4, L'Hôpital's rule
lim x→0 sin(x) / x
1
- 1
Confirm 0/0
sin(0)/0 = 0/0 indeterminate - 2
Differentiate
d/dx(sin x) = cos x, d/dx(x) = 1 - 3
New limit
lim x→0 cos(x)/1 = cos(0) = 1
0/0 form, differentiate top and bottom separately.
Example 5, One-sided limit
lim x→0⁺ 1/x
+∞
- 1
Consider x > 0 approaching 0
1/x grows without bound - 2
Compare to left side
From left (x < 0), 1/x → −∞ - 3
Conclusion
Right limit = +∞, two-sided limit does not exist
Approaching from the right side only.
Free limit calculator with step-by-step solutions
Limits are the gateway to calculus, every derivative and every definite integral is defined in terms of a limit. This calculator helps you evaluate limits with the full reasoning shown. The worked examples cover the five most common limit types: direct substitution, factor-and-cancel, limits at infinity, L'Hôpital's rule, and one-sided limits.
When direct substitution works
If a function is continuous at the target point, the limit equals the function value, just plug in. This is the easiest case and covers polynomials, exponentials, sines, cosines at all points in their domain. If substitution gives a well-defined number, you're done.
Indeterminate forms and how to resolve them
Common indeterminate forms include 0/0, ∞/∞, 0·∞, and ∞ − ∞. They don't have a fixed value, the actual limit depends on which functions are producing them. Tools to resolve: factoring and cancelation, rationalizing (multiply by conjugate), and L'Hôpital's rule. The goal is to transform the expression into something where substitution works.
L'Hôpital's rule, the calculus shortcut
If lim f/g gives 0/0 or ∞/∞, then lim f/g = lim f'/g' (provided the second limit exists). Differentiate the numerator and denominator separately (NOT using the quotient rule), then try the limit again. Apply repeatedly if needed. The famous lim x→0 sin(x)/x = 1 is the textbook example.
Limits at infinity and one-sided limits
For lim x→∞, divide top and bottom by the highest power of x to see what dominates. For one-sided limits (x → a⁺ or x → a⁻), consider only values approaching from the specified side, this matters for functions with jumps, vertical asymptotes, or piecewise definitions. The two-sided limit exists only if both one-sided limits agree.
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