Limits

[floral night]

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[whole gyro]

For the first one tan(x)=x + x^3/3 +o(x^3) so what is cot(x)?
It’s 1/tan(x)=1/(x+x^3/3 +o(x^3))=1/x * (1/1+x^2/3 +o(x^2))= 1/x *(1-x^2/3+o(x^2))

Or I think you could also put on the same denominator you’ll have (x-tan(x)/xtan(x)using l’hôpital’s may work

For the second one you can consider 1/x ln(e^2x+x) and use Taylor expansion here as well

[spring brook]

i don't think i know the Taylor thing here

[whole gyro]

Hmmm okay well for the second one I think you can use the definition of the derivative consider the function f(x)=ln(e^2x+x)

well f(0)=ln(1)=0

So lim x—>0 ln(e^2x+x)/x =limx—>0 (f(x)-f(0))/(x-0)=f’(0) but f’(x)=(2e^(2x)+1)/(e^(2x)+x) thus f’(0)=3

Or you could use l’hôpital’s

[spring brook]

that one was much more understandable

thanks sir

[whole gyro]

You’re welcome now apply the exponential function and the limit should be e^3 for the second one

[spring brook]

okay i see

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