## DRico Group Title would lim(x approches infinity) (sqrt(x^2+1))/(2x+1) be DNE since it would basically be infinity/infinity? one year ago one year ago

1. klimenkov Group Title

What is DNE?

2. DRico Group Title

does not exist

3. redham Group Title

That is indeterminate form, so you can't say it doesn't exist. You will have to manipulate it to use L'hopital's

4. klimenkov Group Title

$\lim_{x\rightarrow\infty}\frac{\sqrt{x^2+1}}{2x+1}=\lim_{x\rightarrow\infty}\frac{\sqrt{\frac {x^2}{x^2}+\frac1{x^2}}}{\frac{2x}x+\frac1x}=\lim_{x\rightarrow\infty}\frac{\sqrt{1}}{2}=\frac12$

5. klimenkov Group Title

@redham L'hopital's is not the best idea.

6. Valpey Group Title

No, you can map a limit here.$\lim_{x \rightarrow \infty}\Big(\frac{\sqrt{x^2+1}}{2x+1}\Big)=\frac{\lim_{x \rightarrow \infty}\Big(\frac{d\sqrt{x^2+1}}{dx}\Big)}{\lim_{x \rightarrow \infty}\Big(\frac{d(2x+1)}{dx}\Big)}$

7. DRico Group Title

how do i simplify to sqrt(1)/2

8. klimenkov Group Title

$\lim_{x\rightarrow\infty}\frac{\sqrt{\frac {x^2}{x^2}+\frac1{x^2}}}{\frac{2x}x+\frac1x}=\lim_{x\rightarrow\infty}\frac{\sqrt{1+1/x^2}}{2+1/x}$ $$1/x^2$$ and $$1/x$$ are almost 0 when x goes to infinity. so only 1in numerator and 2 in denominator are left.

9. redham Group Title

@klimenkov I was actually thinking of squaring the denominator under the radical and then using L'Hopital's. Not from the start

10. DRico Group Title

thnaks to all of you

11. DRico Group Title

one more if y'all are still here how to calculate the derivative of f(x)=sinx

12. DRico Group Title

i know the derivative of sinx is cosx but i dont know how to prove it with the derivative funtion

13. klimenkov Group Title

$\lim_{\Delta x\rightarrow0}\frac{\sin(x+\Delta x)-\sin x}{\Delta x}=\lim_{\Delta x\rightarrow0}\frac{\sin x \cos\Delta x+\cos x\sin \Delta x-\sin x}{\Delta x}=$$\lim_{\Delta x\rightarrow0}\frac{\sin x (\cos\Delta x-1)}{\Delta x}+\lim_{\Delta x\rightarrow0}\frac{\cos x\sin \Delta x}{\Delta x}=\cos x$ Try to get it. Why is it so? Why does the first limit equals to 0 and why does the second equals to cos x?

14. DRico Group Title

im not exactly sure...

15. Valpey Group Title