Limits · Definition

Special trigonometric limits

calculuslimitstrigonometry

Before you start you need to know

Two limits with sine and cosine that help solve many other trigonometric limits.

Definition

Two basic limits in radians: \(\frac{\sin x}{x}\) tends to \(1\) and \(\frac{1-\cos x}{x}\) tends to \(0\) as \(x\) tends to \(0\).

\[\begin{gathered}\lim_{x\to0}\frac{\sin x}{x}=1\\ \lim_{x\to0}\frac{1-\cos x}{x}=0\end{gathered}\]

A few limits are proved once (with the squeeze theorem and geometry) and reused constantly. The most famous is \(\lim_{x\to0}\frac{\sin x}{x}=1\), together with its relative \(\lim_{x\to0}\frac{1-\cos x}{x}=0\).

Their importance is that the derivatives of sine and cosine come from them. Without this limit there would be no formula for the derivative of \(\sin x\).

A common trick: for \(\lim_{x\to0}\frac{\sin(kx)}{x}\) multiply and divide by \(k\) to recognize \(\frac{\sin u}{u}\) with \(u=kx\).

Think about it

What is \(\lim_{x\to0}\frac{\sin 3x}{x}\)?

It is 3: rewrite it as \(3\cdot\frac{\sin 3x}{3x}\) and the factor tends to 1.

Exercises 11

Compute the following limits

  1. \(\displaystyle \lim_{x\to 0}\,\frac{\sin {3 x}}{x}\)
  2. \(\displaystyle \lim_{x\to 0}\,\frac{\sin {5 x}}{2 x}\)
  3. \(\displaystyle \lim_{x\to 0}\,\frac{\tan {4 x}}{x}\)
  4. \(\displaystyle \lim_{x\to 0}\,\frac{1 - \cos {x}}{x^{2}}\)
  5. \(\displaystyle \lim_{x\to 0}\,\frac{\sin {2 x}}{\sin {3 x}}\)
  6. \(\displaystyle \lim_{x\to 0}\,\frac{\sin {7x}}{x}\)
  7. \(\displaystyle \lim_{x\to 0}\,\frac{\sin {4x}}{8x}\)
  8. \(\displaystyle \lim_{x\to 0}\,\frac{1-\cos {2x}}{x^{2}}\)
  9. \(\displaystyle \lim_{x\to 0}\,\frac{x}{\sin {3x}}\)
  10. \(\displaystyle \lim_{x\to 0}\,\frac{\tan {2x}}{\sin {5x}}\)
  11. \(\displaystyle \lim_{x\to 0}\,\frac{\sin {x^{2}}}{1-\cos {x}}\)

References

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