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Posted by on Thursday, April 26, 2012 at 10:33pm.

Evaluate the following limits after identifying the indeterminate form. Use Hospital's rule.

d) lim x_0+ (xe^(2x) + 1)^(5/x)


e) lim x_(Pi/2)+ (1 + sec3x)^(cot3x)


Thank you!

  • Calculus 1 - , Friday, April 27, 2012 at 11:27am

    Consider (xe^(2x) + 1)^(5/x) → 1oo

    Now, ln(x) is continuous for x>0, so use that to lower the complexity

    ln(xe^(2x) + 1)^(5/x) = 5/x * ln(xe^(2x) + 1)

    That's now oo * 0, which we can change to 0/0 by writing it as

    ln(xe^(2x) + 1)/(x/5)

    Now we can apply the Rule, to get

    [e^2x (2x+1)/(e^2x+1)] / (1/5) = 5

    Now, that means ln(limit) = 5, and the limit is e^5

    Use the same trick to change

    (1 + sec3x)^(cot3x) → oo0

    to get the limit = 1

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