SOLUTION: Could you help with another limit pleease? limit as theta tends to 0 of theta tan theta / 1 - cos theta

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Question 190131: Could you help with another limit pleease? limit as theta tends to 0 of
theta tan theta / 1 - cos theta
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
... Start with the given limit


Evaluate the limit by plugging in 0 for each


Evaluate the tangent of 0 to get 0. Evaluate the cosine of 0 to get 1.


Multiply and simplify


So


Since is an indeterminate form, this means that we must use L'Hospital's Rule to find the limit. Remember, L'Hospital's Rule states that:

If functions f and g are in some indeterminate form, then




So let's use L'Hospital's Rule to find the limit:

... Start with the original limit


... Derive the numerator and denominator separately



If you evaluate the limit for the last expression, you'll find that



So we must use L'Hospital's again:


... Start with the previous expression


... Derive the numerator and denominator separately


... Combine like terms


... Plug in to evaluate the limit


Multiply and simplify


Evaluate the cosine of 0 to get 1. Evaluate the secant squared of 0 to get 1/1 or just 1


Multiply and reduce


So


Note: you can graph to visually confirm that the limit is 2.