Limits · Definition

Squeeze theorem

calculuslimits

Before you start you need to know

If a function is squeezed between two others with the same limit, its limit is that same value.

Definition

If \(g(x)\le f(x)\le h(x)\) near \(a\) and \(g\) and \(h\) both have limit \(L\) at \(a\), then \(f\) also has limit \(L\) at \(a\).

\[g\le f\le h,\ \lim_{x\to a}g=\lim_{x\to a}h=L\ \Rightarrow\ \lim_{x\to a}f=L\]

If a function is "trapped" between two others with the same limit, it has no choice but to go to the same place. That is the squeeze (sandwich) theorem.

It is mostly used for oscillating functions such as \(x^2\sin(1/x)\): there is no limit "by hand", but \(-x^2\le x^2\sin(1/x)\le x^2\) and both bounds tend to 0.

The practical difficulty is finding the right bounds: they often come from bounding sine or cosine between −1 and 1. Typical error: finding the limit of \(\sin(1/x)\) alone, which does not exist, instead of bounding the whole product. It is also known as the intercalation theorem.

Think about it

Which bounds are used to compute the limit of \(x^2\sin(1/x)\) at 0?

Use \(-1\le\sin(1/x)\le1\), giving \(-x^2\le x^2\sin(1/x)\le x^2\); both bounds tend to 0.

Exercises 10

  1. What is \(\lim_{x\to 0} x^{2}\sin\!\left(\dfrac{1}{x}\right)\)?
  2. If \(1-x^{2}\le f(x)\le 1+x^{2}\) for all x near 0, what is \(\lim_{x\to 0} f(x)\)?
  3. What hypothesis does the squeeze theorem need to conclude that \(\lim_{x\to a} f(x)=L\)?
  4. What is \(\lim_{x\to 0} x\cos\!\left(\dfrac{1}{x}\right)\)?
  5. If \(g(x)\le f(x)\le h(x)\) near \(a\), \(\lim_{x\to a} g=5\) and \(\lim_{x\to a} h=5\), what is \(\lim_{x\to a} f(x)\)?
  6. What is \(\lim_{x\to\infty}\dfrac{\sin x}{x}\)?
  7. Let \(\lfloor y\rfloor\) be the greatest integer less than or equal to \(y\). What is \(\lim_{x\to 0^{+}} x\lfloor 1/x\rfloor\)?
  8. Let \(f(x)=x^{2}\sin(1/x)\) for \(x\ne 0\) and \(f(0)=0\). What is \(f'(0)=\lim_{h\to 0}\dfrac{f(h)-f(0)}{h}\)?
  9. What other name does the squeeze theorem have?
  10. What is \(\lim_{x\to\infty}\dfrac{\cos x+2}{x}\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.