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Calculus/Calculus I (Indeterminate Forms and L'Hospital's Rule)

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Expert: - 6/20/2008


Question
I hope you can help! I am taking a college first year calculus correspondence class. I have am confused with the following three problems and am really hoping to find out how to work limit problems like these! If you could explain how to obtain the correct answer it would be greatly greatly appreciated!!!  

Directions: Find the limit. Use L'Hospital's Rule where appropraite. If there is a more elementary method, consider using it. If L'Hospital's Rule doesn't apply, explain why.

Q1: The limit as x approaches infinity of lnlnx divided by x
Q2: The limit as x approaches negative infinity of x^2 e^x
Q3: The limit as x approaches 0 from the right of (tan2x)^x

On Q1 I'm pretty sure that lnlnx/x becomes (1/xlnx)/1 but I'm stuck after that point. I hope you can help with these!!! Thanks.

Answer
Q1: Yes, you are correct -  It's (1/(xln(x)))/1.  Let x approach infinity on the top and bottom and you'll get the answer.

Q2: L'Hospital's rule could be applied to x^2/e^(-x) two times get the answer.  The two to look at are d^2(x^2)/d^2 and d^2(e^(-x))/dx^2 (the second derivative of the top and bottom).

Q3: Take exp(ln(tan(2x)^x)).  Note that this can be converted to
exp(x*ln(tan(2x)).  Further conversion can be used to say that the function is exp(x/ln(ctn(x)) since cnt^-1(x) = tan(x).

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