Real Limits

[wind goblet]

Anyone there to help solve these limits with little o notation or any other way that doesn't involve Hopital's?

[tacit seal]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[wind goblet]

I also have this one

Not sure how to proceed, trying different approaches

[brave quest]

For the first one, is there a way that you could rewrite the fraction so it matched the definition of the derivative? For the second and third, what is the limit of the denominator for each one?

[wise solstice]

i assume its supposed to say cbrt(2+x) on the second one bottom?

[brave quest]

Example of a limit being rewritten, using the definition of the derivative: $\lim{x\to 3}\frac{e^x - e^3}{\ln(x) - \ln(3)}=\lim{x\to 3}\frac{e^x-e^3}{x - 3}\cdot \frac{x-3}{\ln(x)-\ln(3)}=e^3\cdot 3$

[wise solstice]

$\lim_{x\to 3}\frac{e^x - e^3}{\ln(x) - \ln(3)}=\lim_{x\to 3}\frac{e^x-e^3}{x - 3}\cdot \frac{x-3}{\ln(x)-\ln(3)}=e^3\cdot \frac{1}{3}$

[timid fog]

cute rizzly bear (nom nom nom)

[wise solstice]

i hope he didnt go trying to find out what was wrong

[brave quest]

Guess it should be times 3 at the end, woops

[wise solstice]

for the first, i think you can just rationalize top and bottom
sqrt(x) - sqrt(2) = (x-2)/(sqrt(x) + sqrt(2)), though this will be pretty much the same as the strategy above

[timid fog]

MicMac

[brave quest]

Additionally, has your class talked at all about l'hopital's rule?

[wind goblet]

no since we're using taylor's series with little-o

however these are the basic ones

so they don't involve much of the series

like e^x would at most be 1 + x + o(x)

[brave quest]

Ah, I see. Then wouldn't it be like this: $\lim_{x\to2}\frac{\sqrt{x}-\sqrt{2}}{\sqrt[3]{x}-\sqrt[3]{2}}=\lim_{x\to2}\frac{\sqrt{2} + \frac{1}{2\sqrt{2}}(x-2) + o(x-2)-\sqrt{2}}{\sqrt[3]{2} + \frac{1}{3\cdot 2^{2/3}}(x-2) + o(x-2)-\sqrt[3]{2}} = \lim_{x\to2}\frac{\frac{1}{2\sqrt{2}} + \frac{o(x-2)}{x-2}}{\frac{1}{3\cdot 2^{2/3}} + \frac{o(x-2)}{x-2}}$ And since $\frac{o(x-2)}{x-2}\to 0$ the whole limit has to equal $\frac{\frac{1}{2\sqrt{2}}}{\frac{1}{3\cdot2^{2/3}}}$

[timid fog]

MicMac

[wind goblet]

Thanks

[cinder grove]

@wind goblet has given 1 rep to @brave quest

[wind goblet]

+close