Please help me on a calculus limit problem. thanks?

Question

Find the limit L. Then find (small delta symbol) >0 such that |f(x)-L| < 0.01 whenever < | x-c| < (small delta symbol)

Lim(x^2+4)
x--> 5

if you can explain that would be awesome. thanks in advance.

Answer 1

The limit is 29. For ε = .01, δ=.0001 is more than sufficient, although you can use any δ such that δ < √25.01 - 5 ≈.0009999 .

Basically, the limit of a function at some point is a number that the function gets arbitrarily close to as the value of x gets closer to the function itself - that is, no matter what ε you choose (privded it is greater than zero), you can always find some neighborhood of c such that every value of the function in that neighborhood (except possibly c itself) is closer to the limit than ε. The variable δ specifies how far from c this negihborhood extends. For your problem, you were asked to evaluate the limit at 5 and then find some neighborhood of 5 that fits the above criterion when ε =.01 . Evaluating the limit is easy, since x²+4 is a continuous function, and for a continuous function, the limit of the function at c is equal to the value of the function at c (this is in fact the definition of a continuous function). So just plug 5 into the function to get 29. For the second part of the problem, consider how far from 5 you can stray and still get values between 28.99 and 29.01. Clearly, upper bound poses the tighter restriction, since x will be multiplied by a lightly higher number in this case (namely, the slightly higher value of x). Therefore, any number that you can add to 5, such that f(5) < 29.01, will do. That is why δ=.0001 is more than sufficient, since 5.0001²+4 = 29.00100001 < 29.01. The maximum comes from solving the inequality f(5+δ) < 29.01, since this yields (5+δ)²+4 < 29.01 → (5+δ)² < 25.01 → 5+δ < √25.01 (only positive solutions need be considered, since we want solutions that are close to 5, and negative numbers aren't) → δ < √2501 - 5.

I hope this is detailed enough for you.

Answer 2

Looks like the fundamental theorem of calculus - please review your calculus book before you go on myspace or facebook.

Answer 3

http://www.math.ucdavis.edu/~kouba/ProblemsList.html

http://tutorial.math.lamar.edu/AllBrowsers/2413/TheLimit.asp

check this websites,it should help