Limits ..Graphing & solving them

[qwerty1]

I am not able to solve some questions so I decided to post them here. I got answers to some of them and I just want you to check them.

But some others I don't understand at all.

 

NOTE : PLEASE DO THE QUESTIONS ONLY IF YOU ARE 100% SURE

 

Here are the questions.

 

#1 - #5 ** ANSWER PROVIDED CHECK IT PLEASE **

 

1)

Suppose that x^2-6x-3 <= f(x) <= sqrt(9-x^2) on

[0,3]. Give upper and lower bounds for lim x->0+ f(x) and lim x->3^- f(x)

 

lim (x^2 - 6x - 3) = -3 as x --> 0+

lim sqrt(9 - x^2) = 3 as x -->0+

 

-3 = f(x) = 3

 

lim (x^2 - 6x - 3) = -12 as x --> 3-

lim sqrt(9 - x^2) --> 0 as x --> 3-

 

-12 = f(x) = 0

 

(What is the upper bound and lower bound ? I don't know that.)

 

2)

Does there exist a real number that is exactly one more than its own square root? Justify your answer (hint: intermediate value theorem)

 

x = (sqrt(x)) + 1?

 

f(x) = x - (sqrt x) - 1 > 0?

f(0) = -1

f(4) = 1

 

There is a value a between 0 and 4, where a = (sqrt a) + 1.

Namely, a ˜ 2.618.

 

3)

True or False: if f is continuous at c, then 1/f is continuous at c. Justify your answer with a proof or counterexample.

 

If the lim exists for f at c, then the lim exists for 1/f at c also.

 

(I know the sentence but needed a counter example.)

 

4)

Where is the function

f(x) = | ln(x)+(sin(e^x)/x^2+5) |

defined, and where is it continuous ?

 

f(x) is defined for all x > 0, and is continuous on its domain.

 

(I know the sentence but needed an example.)

 

5)

Determine where the function f(x) = (x^2+5x+6)/(x+3) is discontinuous, and define a continuous extension g of f.

 

I know the function f(x) is discontinuous at x = -2, but I don't know how to define a continuous extension g of f.

 

6) Evaluate

go to this link to see the function because it is confusing if i write it out. The "input" is the function.

 

http://tinyurl.com/3uutnv5

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There are two more graph questions

 

1) Sketch the graph of a function f that satisfies all the following conditions:

 

f(0) = 1

f(-1)= -2

lim x->0 f(x) = inf

lim x->inf f(x) = 0

lim x-> -inf f(x) = 2

lim x-> -1^+ f(x) = 3

lim x-> -1^- f(x) = 1

 

I have the graph on the following link. Check it and tell me what is wrong.

 

http://tinyurl.com/3mzb7fb

 

2) Sketch the graph of a function that has vertical asymptotes at x = -3 and x = 2, horizontal asymptotes at y = 1 and y = 0, and domain (-inf,-3) U (-3,inf)

 

( I have no idea about this one)

 

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I appreciate the help a lot. THANKS

[DrRocket]

This appears to be a duplicate of your thread in the homework section.

 

We will not do your work for you.

 

Further in that thread you state an urgency that indicates that this is work for credit.

 

I will not help you cheat.