Limits Involving Trig and Inverse Trig Functions
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When we are asked to determine a limit involving trig functions, the best strategy is always to try L'Hôpital's Rule. However, this rule is usually not covered until second semester calculus. So, to evaluate trig limits without L'Hôpital's Rule, we use the following identities.
\(\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = \lim_{\theta \to 0}{\frac{\theta}{\sin(\theta)}} = 1 }\) |
\(\displaystyle{ \lim_{\theta \to 0}{\frac{1-\cos(\theta)}{\theta}} = \lim_{\theta \to 0}{\frac{\cos(\theta)-1}{\theta}} = 0 }\) |
Look over both of those limits carefully. Notice what \(\theta\) goes to. Also notice that the expression in the denominator must match the expression within the trig functions. So, for example, if you have \( \sin(3\theta)\) in the first limit, the denominator must also be \(3\theta\).
Steps To Evaluate Trig Limits |
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Step 1 [ direct substitution ] - - directly substitute the variable into the trig function; if you get an indeterminate form, more work is required; if you don't, you are done |
Step 2A [ algebra ] - - if you have an indeterminate form from direct substitution, use algebra to try to get your limit into a form that matches one or both identities above |
Step 2B [ trig identities ] - - if you can't get your limit to match one of the identities above, use trig identities to get your limit into another form; you may be able to get cancellation or you may be able to match one or both of the identities above |
Step 3 [ keep trying ] - - use direct substitution again to see if you no longer have an indeterminate form; you may need to use the Multiplication Rule when evaluating; if you still have an indeterminate form, don't give up; keep working with it until you get it |