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Calculus

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How do I solve lim(x-->pi/4)(sinx-cosx)/cos(2x)
The (x-->pi/4) goes below the lim.
Should i just substitute the x's with pi/4??

  • Calculus -

    This question is related to limits.

  • Calculus -

    If you do that, you will get 0/0. You have to take the ratio of the derivatives.

    lim(x->pi/4)(sinx-cosx)/cos(2x)
    = lim(x->pi/4)(cosx+sinx)/-2 sin(2x)
    = sqrt 2/(-2) = -0.707

  • Calculus -

    That's called using L'Hopital's rule for indeterminate-ratio limits. It works for 0/0 or infinity/infinity. Try it with a number close to pi/4, such as 0.785, and you will see that it gives the right answer.

  • Calculus -

    if you put in PI/4 in to the x, you will get an indeterminate form 0/0.

    Lim (sinx/cos2x - cosx/cos2x)
    multiply numerator and denominator by (sinx+cosx)

    lim (sin^2x -cos^2x)/cos2x(sinx+cosx)

    lim (sin^2x-cos^2x)/(sin^2x-cos^2x)(sinx+cosx)

    lim 1/(sinx+cosx)= 1/sqrt2

    check that.

  • Calculus -

    looking at drwls answer, using L'Hopitals rule (I assumed you had not gotten to it yet), I see I must have made a sign error, you can find it.

  • Calculus -

    Thanks. Actually i did not learn that other rule yet and I don't think we will but thanks anyway it was helpful.
    I solved using bobpursley's idea and got sqrt2/2=1/sqrt2.

  • Calculus -

    Thanks everyone!

  • Calculus -

    I have a similar question relating to limits that I solved
    lim(x-->0) (1-cosx)/2x^2
    I multiplied the numerator and denominator by (1+cosx) to get
    lim(x->0) (1-cos^2x)/2x^2(1+cosx)
    = lim(x->0)sin^2x/2x^2(1+cosx)
    the lim(x->0) (sinx/x)^2 would =1
    I then substituted the remaining portion with 0 and got 1/4
    Is this answer right??

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