1. Limits

I really need some help with this problem. This is the entire problem :-

Let f(x) = (x^4) + 4 and let Limit of f(x) when x reaches 2 be L.
We will try to make the best possible statement about the upper bounds of |f(x)-L|. What we mean by this can be explained in the following manner.
(1). 3.1 is an upper bound of |2.62|
(2). 2.65 is an upper bound of |2.62|
(3). 2.75 is an upper bound of |2.62|
(4). 2.6 is an upper bound of |2.62|
Now (4) is not even correct. So we should select one from (1), (2) or (3). If we make the statement (2) then we are automatically making the statement (1) and (3). The best one coming the given statements is (2).
Question : Under the condition 0<|x-2|<0.01, find the best statement among the followings:

(a) 0.322618 is an upper bound of |f(x)-L|
(b) 0.322576 is an upper bound of |f(x)-L|
(c) 0.32241 is an upper bound of |f(x)-L|
(d) 0.322392 is an upper bound of |f(x)-L|

2. Re: Limits

It's hard to understand what you are asking here. Is this problem exactly as you were given it? It seems strangely written. Why use absolute values signs for a specific positive number? |2.62| is just 2.62! And why say "upper bound" for a single number? An "upper bound" is for a set. All you are saying here is that 2.62 is less than the given numbers (which, as you say, is NOT true of 4). Since $\displaystyle f(x)= x^4+ 4$ is a polynomial, it is continuous for all x and, in particular, for x= 2: the limit is $\displaystyle 2^4+ 4= 20$ so that "|f(x)- L|" is just $\displaystyle |x^4+ 4- 20|= |x^4- 16|$. If x can be any number, and I see no restriction on it, there is no upper bound on that.

For the second question, we are given that "0< |x- 2|< 0.01", so that x lies between 1.99 and 2.01. The largest x can be is 2.01 so that largest that $\displaystyle x^4- 16$ can be is $\displaystyle |2.01^2- 16|= 16.32240801- 16= 0.32240801$. It looks to me like three of the given statements are true. What do you mean by "best"?

3. Re: Limits

Yeah the same problems came to my head as well after reading and trying to analyse this problem. For your last question I think they are asking for the closest number to 0.32240801 that is the answer is
(c) 0.32241 is an upper bound of |f(x)-L|
I think what they mean by the 'best' is that. Thank you for your help

4. Re: Limits

Yes, of the numbers given, ".32241" is the least upper bound. (Of all real numbers, which is the meaning usually given to "least upper bound", $\displaystyle 2.01^4- 16= 0.32240801$ is "least upper bound".)