Calculus limit help

[szadeklord]

My teacher handed us out a practice sheet with some Limit problems on it, and I am having trouble doing them. We just started the class, and am having trouble understanding the concepts.

Find the limit(if it exists).

1. lim x->2

5x - 3
2x + 9


2.lim t->0

t^2 + 1
t


3.lim x->3^-

x²- 9
x - 3


4.lim x->0.5

2x - 1
6x - 3


5.lim x->0

[1/(x-4)] - (1/4)
x


for this one ill write d = the delta symbol

6.lim dx->0

1 - (x + dx)^2 - (1 - x^2)
dx

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied.

7.f(x) =

x + 2
x


8.f(x) =

x + 1
2x + 2

9.f(x) =
{ x, x< 0
x², x>0

 

[Deleted member 4993]

My teacher handed us out a practice sheet with some Limit problems on it, and I am having trouble doing them. We just started the class, and am having trouble understanding the concepts.

Find the limit(if it exists).

1. lim x->2

5x - 3
2x + 9


2.lim t->0

t^2 + 1
t


3.lim x->3^-

x²- 9
x - 3


4.lim x->0.5

2x - 1
6x - 3


5.lim x->0

[1/(x-4)] - (1/4)
x


for this one ill write d = the delta symbol

6.lim dx->0

1 - (x + dx)^2 - (1 - x^2)
dx

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied.

7.f(x) =

x + 2
x


8.f(x) =

x + 1
2x + 2

9.f(x) =
{ x, x< 0
x², x>0

Are you saying that you cannot even start any of these problems?

Please read the post titled "Read before Posting".

We can help - we only help after you have shown your work - or ask a specific question (not a statement like "Don't know any of these")

Please share your work with us indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[mmm4444bot]

These boards are for tutoring; we do not teach classroom material here.

You seemed to have missed reading our guidelines. Please do not post lists of exercises with no efforts shown or questions asked of your own.

Frankly, if you cannot do the first one, something as gone terribly wrong with your studies. It's obvious that the limit is the same as the function value at x = 2.

I'm not trying to be a meanie, but nobody can do these types of exercises without knowing what a limit actually is. Understanding the concept of a limit is where it seems that you need to start.

Anyway, tell us what you've tried or thought about. Where are you stuck? Can you design some specific questions for us?

People come here and demonstrate their effort; volunteers respond with guidance, based on what they see.

Also, please start a new thread for each new exercise.

Cheers :cool:

 

[HallsofIvy]

Apparently your teacher expects you to know some basic "laws of limits":
1) if \(\displaystyle \lim_{x\to a} f(x)= A\) and \(\displaystyle \lim_{x\to b} g(x)= B\) then \(\displaystyle \lim_{x\to a} (f+ g)(x)= A+ B\).
2) if \(\displaystyle \lim_{x\to a} f(x)= A\) and \(\displaystyle \lim_{x\to b} g(x)= B\) then \(\displaystyle \lim_{x\to a} fg(x)= AB\).
3) if \(\displaystyle \lim_{x\to a} f(x)= A\) and \(\displaystyle \lim_{x\to b} g(x)= B\) and \(\displaystyle B\ne 0\) then \(\displaystyle \lim_{x\to a} \frac{f}{g}(x)= \frac{A}{B}\)

In cases where (3) cannot be applied, because \(\displaystyle B= \lim_{x\to a} g(x)= 0\), then either
4) if the numerator does NOT go to 0 the limit does not go to 0, the limit of the fraction does not exist.
5) if the numerator also goes to 0 the limit may still exist- if there is some interval, containing a, such that f(x)= g(x) for all x except x= a, then \(\displaystyle \lim_{x\to a} f(x)= \(\displaystyle \lim_{x\to a} g(x)\)

You might also need what are called the "trivial limits":
a) \(\displaystyle \lim_{x\to a} C= C\) for any constant, C.
b) \(\displaystyle \lim_{x\to a} x= a\).

For example, applying a, b, 1, 2, and 3 to the first problem gives the answer directly. For 2 through 5, you will need 4 and 5.
For 7, do you know the definition of "continuous function"? If not then the obvious first step is to look it up.\)

 

[lookagain]

Apparently your teacher expects you to know some basic "laws of limits":
1) if \(\displaystyle \lim_{x\to a} f(x)= A\) and \(\displaystyle \lim_{x\to b} g(x)= B\) then \(\displaystyle \lim_{x\to a} (f+ g)(x)= A+ B.\)

2) if \(\displaystyle \lim_{x\to a} f(x)= A \) and \(\displaystyle \lim_{x\to b} g(x)= B\) then \(\displaystyle \lim_{x\to a} fg(x)= AB.\)

3) if \(\displaystyle \lim_{x\to a} f(x)= A\) and \(\displaystyle \lim_{x\to b} g(x)= B\) and \(\displaystyle B\ne 0\) then \(\displaystyle \lim_{x\to a} \frac{f}{g}(x)= \frac{A}{B}\)

In cases where (3) cannot be applied, because \(\displaystyle B= \lim_{x\to a} g(x)= 0\),
then either 4) if the numerator does NOT go to 0 the limit does not go to 0, the limit
of the fraction does not exist.
5) if the numerator also goes to 0 the limit may still exist- if there is some interval,
containing a, such that f(x)= g(x) for all x except x= a, then \(\displaystyle \lim_{x\to a} f(x)= \lim_{x\to a} g(x)\)

You might also need what are called the "trivial limits":
a) \(\displaystyle \lim_{x\to a} C = C \ \ for \ \ any \ \ constant, \ \ C.\)

b) \(\displaystyle \lim_{x\to a} x = a.\) Did this equation show up?

For example, applying a, b, 1, 2, and 3 to the first problem gives the answer directly.
For 2 through 5, you will need 4 and 5.

For 7, do you know the definition of "continuous function"?
If not then the obvious first step is to look it up.

HallsofIvy,

in this forum, the [ t e x ] and [/ t e x] pairings ** work instead of the [itex] and [/itex]
pairings.


**Don't have spaces between characters inside each pair of brackets.

 

[HallsofIvy]

Thanks. I should have checked it myself. I have changed all "itex" to "tex", using notepad.