Thread: some limit questions

use the following graph to answer he three questions
(1) which of the statements is valid
(a) lim x->1 f(x)=2
(b)lim x->1 f(x)=0
(c) lim x->1 f(x) doesnot exist

(2) what is lim x->2+ f(x)?
(a)0 (b) 2 (c) limit does not exist

(3) which ,If any, of these limits exists
(a) lim x->2+f(x)
(b) lim x-> 2- f(x)
(c) lim x-> -3 f(x)
d) none of the above

Originally Posted by bobby77

which of the statements is valid
(a) lim x->1 f(x)=2
(b)lim x->1 f(x)=0
(c) lim x->1 f(x) doesnot exist

Take a look at the function as x approaches 1 both from the left and from the right. Does the function approach the same value from each side? The answer is "yes, it approaches the value 0." Note that I am NOT saying that f(0) = 0 (it doesn't.)

So the answer is b).

-Dan

Originally Posted by bobby77

what is lim x->2+ f(x)?
(a)0 (b) 2 (c) limit does not exist

We are looking at what is happening to f(x) as x approaches 2 from the "+" side, or from the right. Since this is a "one-sided" limit, we don't care about what happens from the other side. So what does f(x) approach as we get close to x = 2 from the right? f(x) approaches the value 2. In this case we even have that f(2) = 2.

So the answer is b).

-Dan

Originally Posted by bobby77

which ,If any, of these limits exists
(a) lim x->2+f(x)
(b) lim x-> 2- f(x)
(c) lim x-> -3 f(x)
d) none of the above

Well, we'd better hope that the limit in a) exists because that was problem 2! So yes, a) is an answer.

Looking at the limit of f(x) as x approaches 2 from the "-", or left, side we see that the function value "blows down" (as opposed to blows up) to negative infinity. So this limit does not exist.

There is nothing strange going on at x = -3. It is easy to see that the limit exists on both sides of x = -3 and that f(x) approaches the same value in each case. (f(-3) is about 2 or so). So this limit also exists.

So the answer(s) is a) and c).

-Dan