# Trigonometric limits

### Evaluation of Trigonometric Limits:

Objective: Trigonometric limits involve determining the value that a trigonometric function approaches as the independent variable approaches a specific value or infinity.

Methods for Evaluating Trigonometric Limits:

1. Direct Substitution:

• For simple trigonometric expressions, direct substitution by substituting the value of $x$ into the function can help evaluate the limit.
• Example: ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}=\frac{\mathrm{sin}\left(0\right)}{0}=1$ (using the fact that ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}=1$
2. Trigonometric Identities:

• Using trigonometric identities like ${\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)=1$ or $\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$ can help simplify expressions.
• Example: ${\mathrm{lim}}_{x\to \frac{\pi }{2}}\frac{\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)}={\mathrm{lim}}_{x\to \frac{\pi }{2}}\frac{\mathrm{cos}\left(x\right)}{\sqrt{1-{\mathrm{cos}}^{2}\left(x\right)}}=\frac{0}{0}$(which is an indeterminate form)
3. L'Hôpital's Rule:

• When trigonometric limits result in indeterminate forms ($\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$), applying L'Hôpital's Rule by finding derivatives of the numerator and denominator can be useful.
• Example: ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{tan}\left(x\right)}{x}={\mathrm{lim}}_{x\to 0}\frac{\frac{d}{dx}\left(\mathrm{tan}\left(x\right)\right)}{\frac{d}{dx}\left(x\right)}={\mathrm{lim}}_{x\to 0}\frac{{\mathrm{sec}}^{2}\left(x\right)}{1}=1$
4. Special Trigonometric Limits:

• Memorizing special limits such as ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}=1$ or ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{tan}\left(x\right)}{x}=1$ can aid in evaluating limits faster.
5. Trigonometric Substitutions:

• In more complex expressions, substitution techniques involving trigonometric identities may be employed to simplify and evaluate limits.
• Example: ${\mathrm{lim}}_{x\to \frac{\pi }{4}}\frac{\mathrm{sin}\left(x\right)-\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)}={\mathrm{lim}}_{x\to \frac{\pi }{4}}\frac{\sqrt{2}\mathrm{sin}\left(x-\frac{\pi }{4}\right)}{\sqrt{2}\mathrm{cos}\left(x-\frac{\pi }{4}\right)}=1$