How do find the limit(x-->1) (1-x+lnx)/(1+cos(pix)) using intermediate forms and

Question

How do find the limit(x-->1) (1-x+lnx)/(1+cos(pix)) using intermediate forms and l'hospitals rule? Can someone do step by step please, i am lost.

Explanation / Answer

before applying the l'hospital rule make sure that the form is 0/0 or inf/inf ...0/0 is here (note that l'hospital rule does not apply for other indeterminate forms.. ) now by lop-ital rule( lim x->a) (y/z) = (lim x->a) (y'/z') where y' = differentiation of y wrt x and z' is the differention of z wrt z .... now keep on doing this until u get the form which is no longer in 0/0 or inf/inf forms.... OK now I solve this problem ..... limit(x-->1) (1-x+lnx)/(1+cos(pix)) = limit(x-->1) (-1+(1/x))/(-pi*sin(pi*x)) .. look still it is in the indeterminate form so applying the lop-ital rule again we get limit(x-->1) (-1/x^2) / (-pi*pi*cos(pi*x)) now it is no longer in 0/0 form so applying the limits we get answer = 1/(pi^2)

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