Question

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!

Answer 1

Consider (xe^(2x) + 1)^(5/x) → 1^oo

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