Trig Limits

oObutterfly-chaserOo · #1 $Old$ September 27th, 2009, 01:20 PM

I'm taking my first Calculus course this year and having a lot of trouble, especially with limits with trig functions in them. I haven't used trig for almost two years now and way out of practice. Unfortunately, I had to work when my teacher had scheduled a help session last week. I was just wondering if anyone here could send a little help my way. I do not expect anyone to do all my work for me. Here is one of the twenty problems i am having trouble with:

Lim x-> 0 (x/tan(x))

Thanks in advance to anyone who can explain this to me.

DeMath · #2 $Old$ September 27th, 2009, 01:32 PM

Quote:

Originally Posted by oObutterfly-chaserOo $View Post$

I'm taking my first Calculus course this year and having a lot of trouble, especially with limits with trig functions in them. I haven't used trig for almost two years now and way out of practice. Unfortunately, I had to work when my teacher had scheduled a help session last week. I was just wondering if anyone here could send a little help my way. I do not expect anyone to do all my work for me. Here is one of the twenty problems i am having trouble with:

Lim x-> 0 (x/tan(x))

Thanks in advance to anyone who can explain this to me.

Hint:

$\mathop {\lim }\limits_{x \to 0} \frac{x} {{\tan x}} = \mathop {\lim }\limits_{x \to 0} \frac{{x\cos x}} {{\sin x}} = \left( {\mathop {\lim }\limits_{x \to 0} \cos x} \right) \cdot {\left( {\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}} {x}} \right)^{ - 1}}$

bgonzal8 · #3 $Old$ September 27th, 2009, 03:18 PM

Quote:

Originally Posted by oObutterfly-chaserOo $View Post$

I'm taking my first Calculus course this year and having a lot of trouble, especially with limits with trig functions in them. I haven't used trig for almost two years now and way out of practice. Unfortunately, I had to work when my teacher had scheduled a help session last week. I was just wondering if anyone here could send a little help my way. I do not expect anyone to do all my work for me. Here is one of the twenty problems i am having trouble with:

Lim x-> 0 (x/tan(x))

Thanks in advance to anyone who can explain this to me.

Hello there. The most important things to realize when dealing with limits are the discontinuities in the function (that should always be your first step in finding the limit), tangent has vertical asymptotes, although this Lim will not deal with the functions asymptote. But remember that your graphing calculator is very handy when dealing with limits.

There are a few methods that you can consecutively try to find the limit.First just try substituting the number x is approacing for the variable in the function ... in this case x-> 0 so Lim x--> (0/tan0) this leads to 0/0 which is no good. Next you can try factoring, but when that won't work here. Another method would be conjugation - where the function is rational with a square root. Okay so for Trig functions you must brush up on your trig Identities. It is really helpful to buy Calculus for Dummies, it got me through calc 1 and now I'm in calc 2 and it's still well worth the 20 $, I would have paid a good 60 $ for the amount its helped me. Anyway tanx = sinx/cosx

Lim x-->0 (x/tanx) = x/sinx/cosx = xcosx/sinx : cosx/sinx = cotx (another trig identity)..... so x(cotx)=0cot0=0; Limx-->0 (x/tanx)=0

mbacarella · #4 $Old$ October 9th, 2009, 05:38 PM

I'm only a first year Calc student but I think the previous post is wrong.

This is how I solved it:

$\lim_{x\to0} \frac{x}{tan(x)}= \lim_{x\to0} \frac{x}{\frac{sin(x)}{cos(x)}}= \lim_{x\to0} \frac{x \times cos(x)}{sin(x)}= \lim_{x\to0} \frac{1 \times cos(x)}{1}= \lim_{x\to0} \frac{1 \times 1}{1}= 1$

Recall:
$tan(x) = \frac{sin(x)}{cos(x)}$
and also
$\lim_{x\to0} \frac{sin (x)}{x} = 1 = \lim_{x\to0} \frac{x}{sin(x)}$

Numerical analysis supports:

Quote:

$ python
Python 2.4.3 (#1, Jul 27 2009, 17:56:30)
[GCC 4.1.2 20080704 (Red Hat 4.1.2-44)] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> from math import tan
>>> f = lambda x: x/(tan(x))
>>> f(0.1)
0.99666444232592377
>>> f(0.01)
0.99996666644444221
>>> f(0.001)
0.99999966666664442
>>> f(-0.001)
0.99999966666664442
>>> f(-0.01)
0.99996666644444221
>>> f(-0.1)
0.99666444232592377

mr fantastic · #5 $Old$ October 9th, 2009, 10:45 PM

Quote:

Originally Posted by bgonzal8 $View Post$

[snip]
Lim x-->0 (x/tanx) = x/sinx/cosx = xcosx/sinx : cosx/sinx = cotx (another trig identity)..... so x(cotx)=0cot0=0; Limx-->0 (x/tanx)=0

Sorry but your calculation is wrong. cot(0) is undefined and so you have an indeterminant form $0 \times \infty$ . At best you have taken a standard indeterminant form 0/0 and made the problem more complicated.