How do I find this limit?

Question

The limit as x approaches pi over 4 of (sinx)-(cosx)/(cos2x). I don't know how to find the derivative of cos2x so that I can use the quotient rule. Help!

I'm not sure if the second person understood the question or not... I wanted the limit as x approaches pi over 4, not the limit as x approaches pi. It's kinda hard to type in math stuff on here.

Answer 1

As you specified that limit as 'x' tends to (pi/4), I have changed the solution accordingly.

Lim x--> (pi/4) [{sin(x) - cos(x)}/cos(2x)]
= Lim x--> (pi/4) [{sin(x) - cos(x)}/{cos^2(x) - sin^2(x)}]
= Lim x--> (pi/4) [{sin(x) - cos(x)}/{(cos(x) + sin(x))*(cos(x) - sin(x))}]
= Lim x--> (pi/4) [-1/{cos(x) + sin(x)}]
= [-1/{cos(pi/4) + sin(pi/4)}]
= -1/[{1/sqrt(2)} + {1/sqrt(2)}]
= -1/[2/sqrt(2)]
= -{sqrt(2)}/2

You don't need to use L.Hospital's rule.

Answer 2

Limit of this function can be found using L,Hosrital rule.

In simple cases, L'Hôpital's rule states that
Formal definition

when determining the limit of a quotient f(x)/g(x) \ when both f and g approach 0, or f and g approach infinity, L'Hôpital's rule states that f'/g' \ has the same limit (if the limit exists), provided that g′ is nonzero throughout some interval containing the point in question. This differentiation often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily.

Herein the given problem, let f(x)= (sinx)-(cosx), and limit of this function as x tends to pi/4 is 0 ; because sin(pi/4)=cos(pi/4).
And, let g(x)=cos(2x), here also as x tends to pi/4, g(x)=cos(2x)=cos(2pi/4)=cos(pi/2)=0.
Consequently,(sinx)-(cosx)/(cos2x)=f(x)/g(x)=0/0 ; which is the in the indeterminate form. Hence L,Hospital rule is applicable.
To apply the rule we have, f ' (x)= (sinx)+(cosx)
And, g ' (x)=-2sin(2x)
Then, limit as xtends to pi/4, f ' (x)/g ' (x) =limit as x tends to pi/4,(sinx)+(cosx)/-2sin(2x) = - quarts2/2= -1/squart 2

Pbhowmic`s solution is also laudable.

Answer 3

If you have learned the chain rule, just use that. If you haven't, the derivative of cos(2x) is -2sin(2x)

Answer 4

generalize that and put it into mathwords.com