October 26th, 2017, 01:42 PM  #1 
Newbie Joined: Oct 2017 From: azerbaijan Posts: 3 Thanks: 0  Limits questions help
Hi i'm kinda having problems with these two questions. I wasn't at the lecture and i don't know how to solve them.

October 26th, 2017, 03:20 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
(15) Notice that the denominator, x 2, does NOT go to 0. So $\displaystyle \lim_{x\to 4}\frac{f(x) 5}{x2}= \frac{\lim_{x\to 4} f(x) \lim_{x\to 4} 5}{\lim_{x\to 4}x \lim_{x\to 4} 2}= 1$. I presume you know that $\displaystyle \lim_{x\to 4} x= 4$, that $\displaystyle \lim_{x\to 4}5= 5$, and that $\displaystyle \lim_{x\to 4} 2= 2$. (16a) Well, basically, you do what the problem tells you to do! Since you are asked to assure that $\displaystyle f(x) L< \epsilon$, start by calculating [math]f(x) L[/math[ for each problem. In the first problem, $\displaystyle f(x)= \sqrt{x}= x^{1/2}$, $\displaystyle L= 1/2$, and $\displaystyle \epsilon= 1/10$ so you need to look at $\displaystyle f(x) L= \sqrt{x} 1/2< 1/10$. Removing the absolute value, $\displaystyle 1/10< \sqrt{x} 1/2< 1/10$, $\displaystyle 1/10+ 1/2= 2/5< \sqrt{x}< 1/10+ 1/2= 3/5$. Now we want to look at $\displaystyle x x_0= x 1/4$. Squaring the previous inequality $\displaystyle 0\le x< 9/24$ so $\displaystyle x 1/4< 9/24 6/24= 3/24$. Each step is "reversible" so if $\displaystyle x 1/4< 3/24$, it will be true that $\displaystyle \sqrt{x} 1/2< 1/10$. Do b and c in the same way. Last edited by Country Boy; October 26th, 2017 at 03:50 PM. 
October 26th, 2017, 10:48 PM  #3 
Newbie Joined: Oct 2017 From: azerbaijan Posts: 3 Thanks: 0 
I can't understand why the squaring of inequality is 0≤x<9/24

October 27th, 2017, 03:57 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
Since the square root of x is a real number, x cannot be negative. Since the square root of x is less than 3/5 x cannot be larger than 9/25. The "24" is a typo. The final result should be 9/25 1/4= 35/100 25/100= 10/100= 1/10 so x 1/4< 1/10. Sorry about that. Last edited by Country Boy; October 27th, 2017 at 04:01 AM. 

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