Calculating Limits: Need Help Solving These!

[n3ll4f]

Hi

I´m new here and i´m new on this course. I have a test tomorrow and i need to know how to calculate limits, but i have some that i can´t solve, please can you solve it, ins´t homework, it´s only to learn (don´t use l'hospital):

lim | x | / (x + 1)
x->0


lim (1 + sin x)^(1/x)
x->0


lim (e^(2 sin x) - e^(sin x)) / (sin 2x)
x->0


Thanks in advance

 

[lurflurf]

lim | x | / (x + 1)
x->0

Show the function is continuous at 0
limit of a continuous function is an evaluation

lim (1 + sin x)^(1/x)
x->0

rewrite as

lim (1 + x[sin x/x])^(1/x)
x->0
use
lim (1 + x*a)^(1/x)=exp(a)
x->0
and composition or squish theorem

lim (e^(2 sin x) - e^(sin x)) / (sin 2x)
x->0

rewrite as
lim (e^(2 y) - e^y) / y
y->sin(x)->0

and

(e^(2 y) - e^y)=(e^y-1)^2+(e^y-1)

and

lim (e^x- 1) / x=1
x->0