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Limits & Continuity · Limit Laws

Limit of a Sum

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

The limit of a sum equals the sum of the limits, provided both limits exist.

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

Worked examples

Find lim(x→2) (x² + 3x).
  1. lim(x→2) x² + lim(x→2) 3x
  2. = 4 + 6

Answer: 10

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Find lim(x→0) (sin x + x).

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Related formulas

Limit of a ProductLimit of a Quotient

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 Sum?

The limit of a sum equals the sum of the limits, provided both limits exist. It is one of the limits & continuity formulas in the CalcRef reference.

When does the Limit of a Sum apply?

The Limit of a Sum holds under this condition: Both lim f(x) and lim g(x) must exist.

How do you use the Limit of a Sum?

Worked example. Find lim(x→2) (x² + 3x). lim(x→2) x² + lim(x→2) 3x. = 4 + 6. Answer: 10.

Practice

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