Applications of derivatives · Definition

L'Hôpital's rule

calculusapplicationslimits

Before you start you need to know

L'Hôpital's rule lets you evaluate indeterminate limits by differentiating numerator and denominator, without simplifying the original expression.

Definition

Rule that computes limits of quotients in indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) by differentiating numerator and denominator separately.

\[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\]

For indeterminate forms \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), L’Hôpital’s rule says you may differentiate numerator and denominator separately: \(\lim\frac{f}{g}=\lim\frac{f'}{g'}\), if the latter exists.

Guide example: in \(\lim_{x\to 0}\frac{\sin x}{x}\) we get 0/0; differentiating gives \(\frac{\cos x}{1}\), and at \(x=0\) the limit is \(1\).

It is not the quotient rule: you do not differentiate the quotient, but each part on its own. And it only applies when the form is really indeterminate.

Think about it

Compute \(\lim_{x\to0}\frac{\sin x}{x}\) with L’Hôpital.

It is 0/0; differentiating, \(\frac{\cos x}{1}\to1\).

Why is the quotient \(f/g\) not differentiated with the quotient rule?

L'Hôpital's rule differentiates numerator and denominator separately; differentiating the whole quotient gives a different expression and changes the limit.

Exercises 20

Compute the following limits

  1. \(\displaystyle \lim_{x\to 0}\,\frac{- x + e^{x} - 1}{x^{2}}\)
  2. \(\displaystyle \lim_{x\to \infty}\,\frac{\ln{\left(x \right)}}{x}\)
  3. \(\displaystyle \lim_{x\to 0}\,\frac{x - \sin {x}}{x^{3}}\)
  4. \(\displaystyle \lim_{x\to 0}\,\frac{e^{2 x} - 1}{3 x}\)
  5. \(\displaystyle \lim_{x\to \infty}\,x^{2} e^{- x}\)
  6. \(\displaystyle \lim_{x\to 1}\,\frac{\ln{\left(x \right)}}{x - 1}\)
  7. \(\displaystyle \lim_{x\to 0}\,\frac{\sin{\left(3 x \right)}}{x}\)
  8. \(\displaystyle \lim_{x\to 2}\,\frac{x^{2} - 4}{x - 2}\)
  9. \(\displaystyle \lim_{x\to 0}\,\frac{1 - \cos{x}}{x^{2}}\)
  10. \(\displaystyle \lim_{x\to 0^{+}}\,x\ln{\left(x \right)}\)
  11. \(\displaystyle \lim_{x\to0}\frac{e^{2x}-1}{x}\)
  12. \(\displaystyle \lim_{x\to2}\frac{x^2-4}{x-2}\)
  13. \(\displaystyle \lim_{x\to0}\frac{1-\cos x}{x^2}\)
  14. \(\displaystyle \lim_{x\to\infty}\frac{\ln x}{x}\)
  15. \(\displaystyle \lim_{x\to\infty}x^2e^{-x}\)
  16. \(\displaystyle \lim_{x\to0}\frac{\sin x-x}{x^3}\)
  17. \(\displaystyle \lim_{x\to0^+}x\ln x\)
  18. \(\displaystyle \lim_{x\to0}\frac{\tan x-x}{x^3}\)

Problems

  1. A student applies L'Hôpital to \(\lim_{x\to0}\frac{x+1}{x+2}\) and gets \(1\). What went wrong?
  2. Let \(\displaystyle L=\lim_{x\to\infty}\frac{x+\sin x}{x}\). With \(f=x+\sin x\), \(g=x\), the quotient is \(\frac{f'}{g'}=1+\cos x\). What is true?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §5.3, Teoremas 5.3.5 y 5.3.6
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 5, Teorema 5.13
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §6.3, Teoremas 6.3.3 y 6.3.4