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Limits & Continuity · Limit Laws

Limit of a Quotient

lim⁡x→af(x)g(x)=lim⁡x→af(x)lim⁡x→ag(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}

The limit of a quotient equals the quotient of the limits, provided the denominator limit is nonzero.

Conditions. Both limits must exist and lim g(x) ≠ 0.
  • 2 worked examples
  • 18 in Limits & Continuity

Worked examples

Find lim(x→2) (x²+1)/(x-1).
  1. lim(x→2)(x²+1) = 5, lim(x→2)(x-1) = 1 ≠ 0
  2. So the limit = 5/1

Answer: 5

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Find lim(x→4) (x-4)/(√x - 2).

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Related formulas

Limit of a SumLimit of a Product

See all 18 Limits & Continuity formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Limit of a Quotient?

The limit of a quotient equals the quotient of the limits, provided the denominator limit is nonzero. It is one of the limits & continuity formulas in the CalcRef reference.

When does the Limit of a Quotient apply?

The Limit of a Quotient holds under this condition: Both limits must exist and lim g(x) ≠ 0.

How do you use the Limit of a Quotient?

Worked example. Find lim(x→2) (x²+1)/(x-1). lim(x→2)(x²+1) = 5, lim(x→2)(x-1) = 1 ≠ 0. So the limit = 5/1. Answer: 5.

Practice

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