Limit Questions: File with Solved Examples and Corrections

[transgalactic]

i added a file in which i tried to solved them

in every limit i solved please correct me where i get it wrong.

because some how my answer differs a lot from the book answer

maybe it some basic knowledge that i lack

please help

 

[lalbatros]

Could not read your first problem.
What is the meaning of lin(x)/x ?

 

[transgalactic]

its S
sinx/x

lim (sinx)/x =1
x>0

 

[lalbatros]

Then the first is easy.
It is of the form lim (f(x))^(sinx/x),
and lim f(x) is easyli obtained.

 

[transgalactic]

so did i get it right??
what about the rest??

 

[VietDao29]

Nope, nope, it's incorrect.
You should note that the limit for e is:
\lim_{x \rightarrow \infty} \left( 1 + \frac{1}{x} \right) ^ x

Or

\lim_{x \rightarrow 0} \left( 1 + x\right) ^ \frac{1}{x}

All the two are of the Indeterminate Form 1 ^ \infty, whereas in your problem, it's not any Indeterminate Forms.

--------------------

\lim_{x \rightarrow 0} \left( \frac{x ^ 2 - 2x + 3}{x ^ 2 - 3x + 2} \right) ^ \frac{\sin x}{x} = \left( \frac{0 ^ 2 - 2 \times 0 + 3}{0 ^ 2 - 3 \times 0 + 2} \right) ^ 1 = \frac{3}{2}

--------------------

The seconds problem you did it all correct, except for the last part, which should read:
= \fbox{\ln} e = 1

--------------------

The third problem, you've differentiated it incorrectly.

(\arctan (2x))' = \frac{2}{1 + 4x ^ 2}
(\sin (3x))' = 3 \cos (3x)

Ok, now, can you complete the three problems on your own? :)