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Limits & Continuity · Limit Laws

Limit of a Power

lim⁡x→a[f(x)]n=[lim⁡x→af(x)]n\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

The limit of a power equals the power of the limit, for any positive integer n.

Conditions. lim f(x) must exist and n is a positive integer.
  • 2 worked examples
  • 18 in Limits & Continuity

Worked examples

Find lim(x→2) (x+1)³.
  1. [lim(x→2)(x+1)]³ = 3³

Answer: 27

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Find lim(x→-1) (x²+2)².

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Common questions

What is the Limit of a Power?

The limit of a power equals the power of the limit, for any positive integer n. It is one of the limits & continuity formulas in the CalcRef reference.

When does the Limit of a Power apply?

The Limit of a Power holds under this condition: lim f(x) must exist and n is a positive integer.

How do you use the Limit of a Power?

Worked example. Find lim(x→2) (x+1)³. [lim(x→2)(x+1)]³ = 3³. Answer: 27.

Practice

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