CalcRef

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.

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

Find lim(x→4) (x-4)/(√x - 2).
  1. Direct substitution gives 0/0, so rationalize: multiply by (√x+2)/(√x+2)
  2. (x-4)(√x+2)/((√x-2)(√x+2)) = (x-4)(√x+2)/(x-4) = √x+2
  3. lim(x→4) (√x+2) = 2+2

Answer: 4

Related formulas

Limit of a SumLimit of a Product

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