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 190130: 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) (Show Source): You can put this solution on YOUR website!
}{1-\cos(\theta)})
... Start with the given limit
}{1-\cos(0)})
Evaluate the limit by plugging in 0 for each 
}{1-1})
Evaluate the tangent of 0 to get 0. Evaluate the cosine of 0 to get 1.
Multiply and simplify
So }{1-\cos(\theta)}=\frac{0}{0})
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
}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)})
So let's use L'Hospital's Rule to find the limit:
... Start with the original limit
+\theta\sec^2(\theta)}{\sin(\theta)})
... Derive the numerator and denominator separately
If you evaluate the limit for the last expression, you'll find that +\theta\sec^2(\theta)}{\sin(\theta)}=\frac{0}{0})
So we must use L'Hospital's again:
... Start with the previous expression
+\sec^2(\theta)+2\theta\sec(\theta)}{\cos(\theta)})
... Derive the numerator and denominator separately
+2\theta\sec(\theta)}{\cos(\theta)})
... Combine like terms
+2\cdot0\sec(0)}{\cos(0)})
... Plug in 
to evaluate the limit
}{\cos(0)})
Multiply and simplify
}{1})
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 }{1-\cos(\theta)}=2)
Note: you can graph to visually confirm that the limit is 2.
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