Limit of a composite function

I don't want an easy answer to this problem. However, I would be happy if you could provide me with theorems and/or techniques required to solve it.

$\displaystyle $\lim _{x\to \infty }x^2(ln({x+1\over x}) +ln({2x+3\over 2x}))$$

I know that $\displaystyle $\lim _{x\to \infty }({x+1\over x})^x = \lim _{x\to \infty }(1+{1\over x})^x = e $$ But the natural logarithm is in the way and I think that you can't calculate the limit IN the logarithm first.

Thanks in advance :)

P.S. Oh, and by the way could anyone recommend a book on calculus with challenging problems, because most of the usual calculus textbooks aren't rigorous enough for my college course. (My lecturer always finds a way to give much more complicated problems than those in textbooks)

Re: Limit of a composite function

Hello, Doubled144314!

We need these two theorems:

. . $\displaystyle \lim_{x\to\infty}\left(1 + \tfrac{1}{x}\right)^x \;=\;e$

. . $\displaystyle \lim_{x\to\infty}\left(1 + \tfrac{a}{x}\right)^x \;=\;e^a$

Quote:

$\displaystyle \displaystyle \lim _{x\to\infty}x^2\bigg[\ln\left(\tfrac{x+1}{x}\right) +\ln\left(\tfrac{2x+3}{2x}\right)\bigg]$

We have: .$\displaystyle \lim_{x\to\infty}x\cdot x\bigg[\ln\left(1+\tfrac{1}{x}\right) + \ln\left(1 + \tfrac{3}{2x}\right)\bigg]$

. . . . . . $\displaystyle =\;\lim_{x\to\infty}x\bigg[x\ln\left(1+\tfrac{1}{x}\right) + x\ln\left(1 + \tfrac{3}{2x}\right)^x\bigg] $

. . . . . . $\displaystyle =\;\lim_{x\to\infty}x\bigg[\ln\left(1 + \tfrac{1}{x}\right)^x + \ln\left(1 + \tfrac{\frac{3}{2}}{x}\right)^x\bigg]$

. . . . . . $\displaystyle =\;\lim_{x\to\infty}x\cdot \bigg[\ln\left(\lim_{x\to\infty}\left[1 + \tfrac{1}{x}\right]^x\right) + \ln\left(\lim_{x\to\infty}\left[1 + \tfrac{\frac{3}{2}}{x}\right]^x\right)\bigg]$

. . . . . . $\displaystyle =\;\infty\cdot \ln(e)\cdot \ln(e^{\frac{3}{2}}) \;=\;\infty\cdot1\cdot\tfrac{3}{2} \;=\;\infty$

Re: Limit of a composite function

Quote:

Originally Posted by

**Doubled144314** I don't want an easy answer to this problem. However, I would be happy if you could provide me with theorems and/or techniques required to solve it.

$\displaystyle $\lim _{x\to \infty }x^2(ln({x+1\over x}) +ln({2x+3\over 2x}))$$

I know that $\displaystyle $\lim _{x\to \infty }({x+1\over x}) = \lim _{x\to \infty }(1+{1\over x})^1 = e $$ But the natural logarithm is in the way and I think that you can't calculate the limit IN the logarithm first.

Thanks in advance :)

P.S. Oh, and by the way could anyone recommend a book on calculus with challenging problems, because most of the usual calculus textbooks aren't rigorous enough for my college course. (My lecturer always finds a way to give much more complicated problems than those in textbooks)

I would rewrite the function as $\displaystyle \displaystyle \begin{align*} \lim_{x \to \infty} \frac{\ln{\left(\frac{x+1}{x}\right)} + \ln{\left(\frac{2x+3}{2x}\right)}}{\frac{1}{x^2}} \end{align*}$, and since this goes to $\displaystyle \displaystyle \begin{align*} \frac{0}{0} \end{align*}$ you can apply L'Hospital's Rule.