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Section 1.5 Limits of Trig Functions

If \(c\) is any number in the natural domain of the function, then the following are true:

  • \(\displaystyle \lim_{x \to c} \sin(x) = \sin(c)\)

  • \(\displaystyle \lim_{x \to c} \csc(x) = \csc(c)\)

  • \(\displaystyle \lim_{x \to c} \cos(x) = \cos(c)\)

  • \(\displaystyle \lim_{x \to c} \sec(x) = \sec(c)\)

  • \(\displaystyle \lim_{x \to c} \tan(x) = \tan(c)\)

  • \(\displaystyle \lim_{x \to c} \cot(x) = \cot(c)\)

Natural Domain.

\(\sin(x)\) has zeros at multiples of \(\pi\text{,}\) so \(\csc(x) = \dfrac{1}{\sin(x)}\) will be undefined at those numbers. Everywhere else is the natural domain (allows us to bypass where the function is discontinuous).

Let's start with limits at infinity.

\begin{align*} \lim_{ x \to \infty } \sin \left( \dfrac{\pi x }{2 - 3x} \right) \amp = \sin \left( \lim_{ x \to \infty } \dfrac{\pi x }{2 - 3x} \right) \\ \amp = \sin \left( \lim_{ x \to \infty } \dfrac{\pi x}{-3x} \right) \\ \amp = \sin \left( \lim_{ x \to \infty } - \dfrac{\pi}{3} \right) \\ \amp = - \sin \left( \dfrac{\pi}{3} \right) \\ \amp = - \dfrac{\sqrt{3}}{2} \end{align*}
General Steps.

  • move limit inside

  • evaluate limit

  • apply trig function to result

There are a couple special limits that we should be aware of:

\begin{equation*} \lim_{x \to 0} \dfrac{\sin(x)}{x} = 1 \mbox{ and } \lim_{x \to 0} \dfrac{1-\cos(x)}{x} = 0 \end{equation*}

Example 1.5.1.
\begin{align*} \lim_{x \to 0} \dfrac{\sin(7x)}{x} \amp = \lim_{x \to 0} \dfrac{\sin(7x)}{x} \cdot \dfrac{7}{7} \\ \amp = \lim_{x \to 0} \dfrac{7(\sin(7x))}{7x} \\ \amp = 7 \left( \lim_{x \to 0} \dfrac{\sin(7x)}{7x} \right) \\ \amp = 7 \left( \lim_{x \to 0} \dfrac{\sin(\theta)}{\theta} \right) \\ \amp = 7 \cdot (1)\\ \amp = 7 \end{align*}