MHB Calculating Limits without Cheating or L'Hopital's Rule

Yankel
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Hello all,

I am trying to calculate the following limits, without cheating and using a calculator (by setting a very close value of the required value of x). And no l'hopital's rule either if possible :-)

The limits are:

\[\lim_{x\rightarrow 0} \frac{ln(x^{2}+e^{x})}{ln(x^{4}+e^{2x})}\]

\[\lim_{x\rightarrow \infty } \frac{ln(x^{2}+e^{x})}{ln(x^{4}+e^{2x})}\]

Thank you.
 
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mmm...after rethinking the problem. If on the denominator, I take the power of 2 up, and then using ln rules takes it out of the ln, I get 0.5. Is this the answer for both cases ?

If so, I apologize, and here is a new question instead :-)

\[\lim_{x\rightarrow \infty }[(x+2)ln(x+2)-2(x+1)ln(x+1)+xln(x)]\]
 
Last edited:
Yankel said:
mmm...after rethinking the problem. If on the denominator, I take the power of 2 up, and then using ln rules takes it out of the ln, I get 0.5. Is this the answer for both cases ?

If so, I apologize, and here is a new question instead :-)

\[\lim_{x\rightarrow \infty }[(x+2)ln(x+2)-2(x+1)ln(x+1)+xln(x)]\]

Hint: use that

$$\lim_{x \to \infty } x \log(x+1)-x \log(x) = 1 $$
 
I'm afraid that the hint is not helping, I don't know how to proceed.

In addition, my solution to the first limits is wrong (final answer is correct, but the way is wrong).
 
The first one (using the limit definition of $e$ and approximation to Taylor series):

$$\lim_{x\to0}\dfrac{\ln(x^2+e^x)}{\ln(x^4+e^{2x})}=\lim_{x\to0}\dfrac{\ln(x^2+[(x+1)^{1/x}]^x)}{\ln(x^4+[(2x+1)^{1/x}]^x)}=\lim_{x\to0}\dfrac{\ln(x^2+x+1)}{\ln(x^4+2x+1)}\sim\dfrac{\ln(e^x)}{\ln(e^{2x})}=\dfrac12$$

Third one:

$$\ln\left(\lim_{x\to\infty}\dfrac{(x+2)^{x+2}\cdot x^x}{(x+1)^{2(x+1)}}\right)\sim\ln(1)=0$$
 
greg1313 said:
The first one (using the limit definition of $e$ and approximation to Taylor series):

$$\lim_{x\to0}\dfrac{\ln(x^2+e^x)}{\ln(x^4+e^{2x})}=\lim_{x\to0}\dfrac{\ln(x^2+[(x+1)^{1/x}]^x)}{\ln(x^4+[(2x+1)^{1/x}]^x)}=\lim_{x\to0}\dfrac{\ln(x^2+x+1)}{\ln(x^4+2x+1)}\sim\dfrac{\ln(e^x)}{\ln(e^{2x})}=\dfrac12$$

This technique (with the limit definition of $e$) is questionable. Consider using it to solve

$$\lim_{x\to0}\dfrac{e^x-x-1}{x^2}$$
 

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