Limits · Definition

One-sided limits

calculuslimits

Before you start you need to know

One-sided limits measure what f does as x approaches a from only one side.

Definition

The left-hand limit uses only values \(x<a\) and the right-hand limit only values \(x>a\). The limit of \(f\) at \(a\) exists if and only if both one-sided limits exist and are equal.

\[\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\]

You can approach \(a\) from two sides: from the left (\(x<a\)) and from the right (\(x>a\)). Each path has its own one-sided limit.

The "full" limit exists only when both agree. That is why, with piecewise functions or absolute values, it pays to compute both sides separately before concluding.

A useful example: \(\frac{|x|}{x}\) equals \(-1\) to the left of 0 and \(1\) to the right. Each one-sided limit exists, but they differ, so the limit does not exist.

Think about it

If the left-hand limit is 3 and the right-hand limit is 5, does the limit exist?

No: the one-sided limits exist but are different.

Exercises 10

Compute the following limits

  1. \(\displaystyle \lim_{x\to 0^-}\,\frac{\left|{x}\right|}{x}\)
  2. \(\displaystyle \lim_{x\to 0^+}\,\frac{\left|{x}\right|}{x}\)
  3. \(\displaystyle \lim_{x\to 2^+}\,\frac{1}{x - 2}\)
  4. \(\displaystyle \lim_{x\to 2^-}\,\frac{1}{x - 2}\)
  5. \(\displaystyle \lim_{x\to 0^-}\,\frac{1}{x^{2}}\)
  6. \(\displaystyle \lim_{x\to 1^-}(2x+1)\)
  7. \(\displaystyle \lim_{x\to 0^+}\,\frac{\sin x}{|x|}\)

Problems

  1. For \(\lim_{x\to a} f(x)\) to exist it is necessary and sufficient that...
  2. Let \(f(x)=x+1\) if \(x<2\) and \(f(x)=x^2-1\) if \(x\ge 2\). Compute \(\lim_{x\to 2} f(x)\).
  3. If \(\lim_{x\to a^-}f(x)=2\) and \(\lim_{x\to a^+}f(x)=5\), does \(\lim_{x\to a}f(x)\) exist?

References

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