Solving the Limit of sin(cos(x))/sec(x) as x->0

[Miike012]

Problem:

Lim sin(cos(x))/sec(x)
x -> 0

The answer in the book is sin(1).. which is obvious... but why do I not arrive at the same answer doing is this way...

Knowing as x approaches zero, sin(x)/x approaches 1...

( Sin(cos(x))/cos(x) )(cos^2(x)) = (1)(1) = 1 = / = sin(1)...?

 

[gb7nash]

Knowing as x approaches zero, sin(x)/x approaches 1...

( Sin(cos(x))/cos(x) )(cos^2(x)) = (1)(1) = 1 = / = sin(1)...?

The bolded part is correct. However, what is cos(x) approaching as x approaches 0?

 

[Miike012]

cos(x) approaches 1..

 

[gb7nash]

Ok, so since cos(x) goes to 1:

$$\lim_{x \to 0} \frac{\sin(\cos(x))}{\cos(x)}$$

is equivalent to:

$$\lim_{u \to 1} \frac{\sin(u)}{u}$$

We can't apply the sin(_)/_ rule here. The term inside of the sine approaches 1.

 

[Miike012]

Thank you.

 

[Miike012]

I wasn't aware that you can't use that rule if the function actually approaches a value.

 

[gb7nash]

I wasn't aware that you can't use that rule if the function actually approaches a value.

$$\lim_{x \to 0}\frac{\sin(x)}{x} = 1$$

x needs to approach 0. If it approaches any other value (or does not exist), you can't use this identity.

 

[Miike012]

that makes sense. Thank you.