One-Sided Limits

Definition. We say the limit of \(f(x)\) is equal to \(L\) as \(x\) approaches \(c\) from the left and write \[ \lim_{x\to c^-}f(x) = L \] if we can make values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\) to be sufficiently close to \(c\) with \(x\) less than \(c\).

Example 1. Evaluate the following limits based on the graph.




  • Evaluate \(\displaystyle\lim_{x\to 2^-} f(x)\)
  • Evaluate \(\displaystyle\lim_{x\to 0^-} f(x)\)
  • Evaluate \(\displaystyle\lim_{x\to (-1)^-} f(x)\)

Definition. We say the limit of \(f(x)\) is equal to \(L\) as \(x\) approaches \(c\) from the right and write \[ \lim_{x\to c^+}f(x) = L \] if we can make values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\) to be sufficiently close to \(c\) with \(x\) greater than \(c\).

Example 2. Evaluate the following limits based on the graph.




  • Evaluate \(\displaystyle\lim_{x\to 2^+} f(x)\)
  • Evaluate \(\displaystyle\lim_{x\to 0^+} f(x)\)
  • Evaluate \(\displaystyle\lim_{x\to (-1)^+} f(x)\)

Two-Sided Limits

Definition. Suppose \(f(x)\) is defined on some open interval that contains \(c\), except possibly at \(c\) itself. Then we write \[ \lim_{x\to c}f(x) = L \] and say “the limit of \(f(x)\) equals \(L\), as \(x\) approaches \(c\)” if we can make values of \(f(x)\) arbitrarily close to \(L\) by restricting \(x\) to be sufficiently close to \(c\) (on either side of \(c\)) but not equal to \(c\).

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

Example 3. Evaluate the following limits based on the graph.




  • Evaluate \(\displaystyle\lim_{x\to 2} f(x)\)
  • Evaluate \(\displaystyle\lim_{x\to 0} f(x)\)
  • Evaluate \(\displaystyle\lim_{x\to (-1)} f(x)\)

Practice Problems.

Evaluate the following limits based on the given graph.
  • \(\displaystyle\lim_{x\to (-1)^-} f(x)=\)
  • \(\displaystyle\lim_{x\to (-1)^+} f(x)=\)
  • \(\displaystyle\lim_{x\to (-1)} f(x)=\)
  • \(f(-1)=\)
  • \(\displaystyle\lim_{x\to 0^-} f(x)=\)
  • \(\displaystyle\lim_{x\to 0^+} f(x)=\)
  • \(\displaystyle\lim_{x\to 0} f(x)=\)
  • \(f(0)=\)
  • \(\displaystyle\lim_{x\to 1^-} f(x)=\)
  • \(\displaystyle\lim_{x\to 1^+} f(x)=\)
  • \(\displaystyle\lim_{x\to 1} f(x)=\)
  • \(f(1)=\)

Limit Involves Infinity

Example 4. Evaluate the following limits based on the graph.

  • \(\displaystyle\lim_{x\to -\infty} f(x)=\)
  • \(\displaystyle\lim_{x\to \infty} f(x)=\)
  • \(\displaystyle\lim_{x\to (-1)^+} f(x)=\)
  • \(\displaystyle\lim_{x\to (-1)^-} f(x)=\)
  • \(\displaystyle\lim_{x\to 1^+} f(x)=\)
  • \(\displaystyle\lim_{x\to 1^-} f(x)=\)

Practice Problems.

Evaluate the following limits based on the given graph.
  • \(\displaystyle\lim_{x\to (-1)^-} f(x)=\)
  • \(\displaystyle\lim_{x\to (-1)^+} f(x)=\)
  • \(\displaystyle\lim_{x\to 2^-} f(x)=\)
  • \(\displaystyle\lim_{x\to 2^+} f(x)=\)
  • \(\displaystyle\lim_{x\to -\infty} f(x)=\)
  • \(\displaystyle\lim_{x\to \infty} f(x)=\)