Calculate limits step-by-step with L'Hospital's rule and algebraic simplification
x^2, (x-1)/(x+1), sqrt(x), abs(x)sin(x), cos(x), tan(x), asin(x)e^x, exp(x), ln(x), log(x,b)infinity, -infinity, pi
A limit describes the value that a function approaches as the input approaches some value. Limits are fundamental to calculus - they're the foundation for derivatives, integrals, and continuity. The notation limx→a f(x) = L means "as x gets closer to a, f(x) gets closer to L."
limx→a f(x) - Approaches from both left and right
Exists only if left and right limits are equal. Example: limx→2 x² = 4
Left: limx→a⁻ f(x) - Approaches from values less than a
Right: limx→a⁺ f(x) - Approaches from values greater than a
Useful for piecewise functions and discontinuities. Example: limx→0⁻ 1/x = -∞
limx→∞ f(x) or limx→-∞ f(x)
Describes end behavior. Example: limx→∞ 1/x = 0
When direct substitution produces these forms, special techniques are required:
Most common. Use factoring or L'Hospital
Use L'Hospital's rule
Rewrite as 0/0 or ∞/∞
Combine fractions or factor
Use e and logarithms
Take logarithm
If lim f(x)/g(x) gives 0/0 or ∞/∞, then:
limx→a f(x)/g(x) = limx→a f'(x)/g'(x)
Take derivatives of numerator and denominator separately (not quotient rule!). Can be applied repeatedly if needed.
limx→0 sin(x)/x = limx→0 cos(x)/1 = cos(0) = 1
Simply plug in the value. Works if function is continuous at that point.
Factor numerator and denominator, cancel common terms.
Example: (x²-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1
Multiply by conjugate to eliminate radicals.
Example: (√(x+1)-1)/x × (√(x+1)+1)/(√(x+1)+1)
For limits at infinity, divide all terms by highest power of x.
Example: limx→∞ (3x²+5)/(2x²-1) → divide by x²
| Limit | Value |
| limx→0 sin(x)/x | 1 |
| limx→0 (1-cos(x))/x | 0 |
| limx→0 (ex-1)/x | 1 |
| limx→∞ (1 + 1/x)x | e |
| limx→∞ (1 + k/x)x | ek |
A function f(x) is continuous at x = a if:
Types of Discontinuities:
Explore more calculus tools for complete mathematical analysis.
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