I don't understand limits!?

Question

I don't remember A LOT about limits with trig. functions. We're reviewing and I'm having so much trouble.

We haven't done L'Hopital's rule, so please don't suggest it. People have tried to help me solve it this way but I just cannot understand it. There needs to be another way to solve these limits.

Please, please help me out?

There are questions like:

Lim x-->0 (tan πx/ x)
Lim x---> infinity (x^2 sin(1/x)
lim x--> 0 (x csc x)

Do I use the quotient rule? Or...? I'm so confused!

Please help me. My HW is due tomorrow and I'm desperate for help!

Thank you SO much!

Answer 1

The first problem Lim x-->0 (tan πx/ x) , I assume it is Lim x-->0 (tan πx)/ x ... because you mention about quotient rule later on. My way of solving these problems is to imagine 0 is a "very small" number, and "infinity" is a "very big" number.
So, what happens if these functions approaching a very small and positive number, or a very big number ?
1) As x--> 0, tan πx --> tan 0 (everything multiplies by 0 is 0), and tangent of 0 is 0.
As x --> 0, x becomes smaller and smaller, but you can't say x=0 because you get undefined number if the denominator is 0. As the denominator approaches 0, the whole number gets bigger and bigger. However the numerator (or the top) is approaching 0 (or 0), the whole thing will eventually becomes 0; because when a 0 is divided by any number, the results is always 0.
So, the answer for problem 1 is 0.

2) Using the same analysis, as x--> infinity, x^2 is a very big, huge number.
as x --> infinity, sin (1/x) --> sin0 (because the inverse of a huge number is always small and eventually goes to 0). Sin0 is 0 . Therefore, a huge number multiplies to a 0, the results is 0.
So, the answer for problem 2 is also 0.
3) As x--> 0, x is very small. csc(x) = 1/(sin x) = 1/small number (again you can't say the denominator is 0 (sin 0=0), because it will give you an undefined number). A small number/ a small number ... you get 1.

I hope this will help you to learn, because limitation is not hard at all.

Answer 2

For the first one, It seems to me you can cancel out the X altogether. Then you are just figuring tan(pi).

For the second one, notice, argument to sine function will approach zero as x gets larger and larger. Sine of zero is zero. Therefore, no matter what you mutiply it with, the result will approach zero.

For the last one, notice, you can rewrite this as
lim x-> 0 (x/sin(x))

At this point, you should be realizing, there is a simple trig limit identity that says this limit equals to ONE.

Answer 3

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Answer 4

Here's a site with some tutorials that might help.

Tutorials for the Calculus Phobe:
http://www.calculus-help.com/funstuff/phobe.html