Lim of trig functions. Does it exist?

[mathgeek69]

1. Does the limit exist of the following:

lim as x→ 1- ((cos^-1(x))/(1-x))



2. Homework Equations :

3. The Attempt at a Solution :

lim as x→ 1- ((cos^-1(x))/(1-x))
= lim as x→ 1- (cos^-1(x))/ lim as x→ 1-(1-x)

Let y = 1-x

lim as y→0 (cos^-1(1-y)) / lim as y→0 (y)
= 0/0 therefore limit of the entire function as x→1- is ∞

 

[SteamKing]

Sorry, 0/0 is an indeterminate form.

 

[mathgeek69]

So 0/0 = Limit doesn't exist ?

 

[SteamKing]

Not necessarily.

For example, the limit of sin(x)/x as x approaches 0 is equal to 1.

 

[verty]

Let $x = cos(y)$, this may help to simplify things.

If not, I would leave this question for later.

 

[Mark44]

So 0/0 = Limit doesn't exist ?

[0/0] is one of several indeterminate forms. The "indeterminate" part means that you can't tell if an expression with this form has a limit, and if it does, what that limit will be.

Some of the other indeterminate forms are [∞/∞], [∞ - ∞], and [1^∞].

All of the limits below are of the [0/0] indeterminate form:
$$ 1. \lim_{x \to 0}\frac{x^2}{x}$$
$$ 2. \lim_{x \to 0}\frac{x}{x^2}$$
$$ 3. \lim_{x \to 0}\frac{x}{x}$$
In #1, the limit is 0; in #2, the limit doesn't exist; in #3, the limit is 1.

 

[eumyang]

[0/0] is one of several indeterminate forms. The "indeterminate" part means that you can't tell if an expression with this form has a limit, and if it does, what that limit will be.

Some of the other indeterminate forms are [∞/∞], [∞ - ∞], and [1^∞].

All of the limits below are of the [0/0] indeterminate form:
$$ 1. \lim_{x \to 0}\frac{x^2}{x}$$
$$ 2. \lim_{x \to 0}\frac{x}{x^2}$$
$$ 3. \lim_{x \to 0}\frac{x}{x}$$
In #1, the limit is 0; in #2, the limit doesn't exist; in #3, the limit is 1.

I think you switched #1 and #2. In #1, the limit doesn't exist, and in #2, the limit is 0.

 

[Mark44]

[0/0] is one of several indeterminate forms. The "indeterminate" part means that you can't tell if an expression with this form has a limit, and if it does, what that limit will be.

Some of the other indeterminate forms are [∞/∞], [∞ - ∞], and [1^∞].

All of the limits below are of the [0/0] indeterminate form:
$$ 1. \lim_{x \to 0}\frac{x^2}{x}$$
$$ 2. \lim_{x \to 0}\frac{x}{x^2}$$
$$ 3. \lim_{x \to 0}\frac{x}{x}$$
In #1, the limit is 0; in #2, the limit doesn't exist; in #3, the limit is 1.

I think you switched #1 and #2. In #1, the limit doesn't exist, and in #2, the limit is 0.

I don't think so. Factor x/x out of each and see what you get.