proving limits by ϵ−δ and ϵ−M

Question

I'm having a real hard time with these two:

1) $$\lim_{x-> ∞} \frac{x}{x+sinx} = 1$$ -- proving only by using "ϵ−M"

2)$$\lim_{x-> 2} \sqrt{3x-2} = 2$$ -- proving only by using "ϵ−δ"

i have tried both a lot of times, tried to find a connection between δ an sqrt{3x-2} without no luck. i got into messy numbers and i know that i'm missing something. if you could show me the way it would help a lot. same for the firt one with the sine. I'm lost.

would really appreciate if you could help me.

Answer 1

For $x>1$ we have

$|\frac{x}{x+sinx} - 1|= \frac{| \sin x |}{x+ \sin x} \le \frac{1}{x+ \sin x} \le \frac{1}{x-1} $.

Your turn !