Thread: Limits and Continuity Help

Can anyone help me out with these 3 calculus problems?

The first one:
f(x) = [1-cos(x)]/(2 sin^2(x))
lim(x-->0)f(x)
f(x) * x/x = [1-cos(x)]/x * x/(2 sin^2(x))
lim(x-->0)[1-cos(x)]/x = 0
lim(x-->0) x/sin x = 1
lim(x-->0) x/sin x = 1
lim(x-->0) 1/(2 sin(x)) = DNE

My teacher told us that the answer is 1/4, but I can't imagine how she got to this point.

For both the 2nd and 3rd- Find the points of discontinuity and label them as removable or nonremovable. I am looking for confirmation, and possibly correction if I am wrong.
The second one:
lim(x--> neg infinity)(5x^4-9x)/(5x^3+9)
Apply direct substitution to get (inf-(neg inf))/(-inf) = -infinity

The third one:
lim(x-->neg inf)sqrt(9x^2+2)/(4x+3)
Apply direct substitution to get (infinity)/(-infinity) = -infinity

Thanks in advance for everyone's help!

Originally Posted by macosxnerd101

Can anyone help me out with these 3 calculus problems?

The first one:
f(x) = [1-cos(x)]/(2 sin^2(x))
lim(x-->0)f(x)
f(x) * x/x = [1-cos(x)]/x * x/(2 sin^2(x))
lim(x-->0)[1-cos(x)]/x = 0
lim(x-->0) x/sin x = 1
lim(x-->0) x/sin x = 1
lim(x-->0) 1/(2 sin(x)) = DNE

My teacher told us that the answer is 1/4, but I can't imagine how she got to this point.








For both the 2nd and 3rd- Find the points of discontinuity and label them as removable or nonremovable. I am looking for confirmation, and possibly correction if I am wrong.
The second one:
lim(x--> neg infinity)(5x^4-9x)/(5x^3+9)
Apply direct substitution to get (inf-(neg inf))/(-inf) = -infinity










The third one:
lim(x-->neg inf)sqrt(9x^2+2)/(4x+3)
Apply direct substitution to get (infinity)/(-infinity) = -infinity

Thanks in advance for everyone's help!

..

Thanks for your help. I'll give #3 a try:

sqrt(9x^2+2)/(4x+3)
[3x/x + sqrt(2)/x] //take 3x out of the radical
divided by (4x/x + 3/x) =
3/4.

Can you confirm this as the right answer?

Originally Posted by macosxnerd101

Thanks for your help. I'll give #3 a try:

sqrt(9x^2+2)/(4x+3)
[3x/x + sqrt(2)/x] //take 3x out of the radical
divided by (4x/x + 3/x) =
3/4.

Can you confirm this as the right answer?

Yup, this is right!

Tonio