Limit Calculator

Evaluate limits of functions numerically as x approaches a specific value.

Function Type:

Coefficient (a):

Power (n):

Coefficient (b):

Constant (c):

Approach Value

x approaches:

Direction:

Limit

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Enter function and approach value

Left-hand Limit

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Right-hand Limit

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f(a)

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Continuity

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Limit Properties

A limit describes the value a function approaches as the input approaches a specific value, even if the function is not defined at that point.

Limit Laws

lim[f(x) + g(x)] = lim f(x) + lim g(x)
lim[f(x) · g(x)] = lim f(x) · lim g(x)
lim[f(x)/g(x)] = lim f(x) / lim g(x) (if lim g(x) ≠ 0)

Important Limits

Limit	Value
lim(x→0) sin(x)/x	1
lim(x→0) (e^x - 1)/x	1
lim(x→0) (1+x)^(1/x)	e
lim(x→∞) (1 + 1/x)^x	e
lim(x→0) ln(1+x)/x	1

L'Hopital's Rule

When lim gives 0/0 or ∞/∞ indeterminate forms:

lim f(x)/g(x) = lim f'(x)/g'(x)

Frequently Asked Questions

When does a limit not exist?

A limit doesn't exist when the left and right limits are different, when the function oscillates infinitely, or when it approaches infinity (vertical asymptote).

What is the difference between limit and function value?

The limit is what the function approaches, while f(a) is the actual value at that point. A function can have a limit even if it's not defined at that point.