Question

Evaluate the following limits after having identified it's indeterminate form:

lim x->0+ (xe^(2x) +1)^(5/x)

My last one and I have no idea how to go about it. I was thinking L'H, but the derivative is long.

Thank you

Answer 1

In google paste:

emathhelp limit calculator

When you see list of results click on:

Limit Calculator - eMathHelp

When page be open in rectangle Enter a function paste (xe^(2x) +1)^(5/x)

In rectangle Find the limit at type 0

In rectangle Choose a direction select Right-hand limit

Then click on CALCULATE

You will see solution step-by-step.

Answer 2

lim x->0+ (xe^(2x) +1)^(5/x) = 1^∞
so, take logs
log(lim) = lim(log) = lim (5/x) log(xe^(2x)+1)
Now we have ∞*0, so if we divide, that becomes
lim log(xe^(2x)+1) / (x/5)
Now L'H yields
= lim 5/(xe^(2x)+1) * e^(2x) (2x+1) = 5
so, now we have lim(log) = 5, so the original limit is e^5