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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

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

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Practice Problems

Instructions - Unless otherwise instructed, evaluate these limits. Give your answers in exact terms.

Level A - Basic

Practice A01




Practice A02



Practice A03



Practice A04




Practice A05



Practice A06



Practice A07



Practice A08



Practice A09



Practice A10



Practice A11



Practice A12



Practice A13



Practice A14



Practice A15



Practice A16



Level B - Intermediate

Practice B01

\(\displaystyle{\lim_{x\to0}{\frac{\cos(2x)-1}{\cos x-1}}}\)



Practice B02




Practice B03



Practice B04



Practice B05



Practice B06



Practice B07



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