Limits · Definition

Limit laws

calculuslimits

Before you start you need to know

The limit laws explain how limits of simple functions combine, so the formal definition does not have to be applied every time.

Definition

If \(\lim_{x\to a} f(x)\) and \(\lim_{x\to a} g(x)\) exist, the limit of their sum, difference, product or quotient (with nonzero denominator) is the same operation applied to the limits.

\[\lim_{x\to a}[f(x)\,g(x)]=\lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x)\]

Computing limits with epsilon and delta every time would be exhausting. The limit laws let you break things down: the limit of a sum is the sum of the limits, the limit of a product is the product, and the limit of a quotient is the quotient as long as the denominator does not tend to 0.

In practice, for "friendly" functions (polynomials, rational functions with nonzero denominator) you can simply substitute \(x=a\). When substitution gives something odd like \(0/0\), the laws are not enough and you must simplify first.

Each law requires the partial limits to exist. Before splitting a limit into pieces, check that every piece converges.

Think about it

What condition does the quotient law need?

That the limit of the denominator is not 0 (and that both limits exist).

Exercises 10

Compute the following limits

  1. \(\displaystyle \lim_{x\to 2}\,\left(x^{2} + 3 x - 1\right)\)
  2. \(\displaystyle \lim_{x\to 4}\,\left(x - 3\right) \left(x + 1\right)\)
  3. \(\displaystyle \lim_{x\to 3}\,\frac{x^{2} + 1}{x - 1}\)
  4. \(\displaystyle \lim_{x\to 4}\,\sqrt{x} \left(x - 1\right)\)
  5. \(\displaystyle \lim_{x\to 2}\,\frac{x^{2}-4}{x-2}\)
  6. \(\displaystyle \lim_{x\to 1}\left[\frac{x^{2}-1}{x-1}+\frac{2}{x+1}\right]\)

Problems

  1. By the product law, what does \(\lim_{x\to a}[f(x)\,g(x)]\) equal if both limits exist?
  2. Besides the existence of the limits of numerator and denominator, what does the quotient law require?
  3. When is it enough to substitute \(x=a\) to obtain the limit of a function?
  4. Can the quotient law be applied to \(\lim_{x\to 3}\frac{x+1}{x-3}\)?

What comes next

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.