If limit of function f(x)=(sin x-cos x)/cos 2x is l, choose the good answer: a)l=0;b)l=-1;c)=6;d)l=1/6;e)l=-square root 2/2
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We have to find the limit of f(x)=(sin x-cos x)/cos 2x for x--> 45 degrees.
We know that cos 2x = (cos x)^2 - (sin x )^2
lim x--> 0 [(sin x-cos x)/cos 2x]
=> lim x--> 0 [(sin x-cos x)/[(cos x)^2 - (sin x )^2]
=> lim x--> 0 [(sin x-cos x)/[(cos x - sin x)(cos x + sin x )]
cancel (sin x - cos x)
=> lim x--> 0 [-1/(cos x + sin x)]
substitute x = 0
=> -1/(cos x + sin x )
=> -1 / (1/sqrt 2 + 1/ sqrt 2)
=> -1/ (2/sqrt 2)
=> -1 / sqrt 2
=> -sqrt 2/2
l = -sqrt 2/2 or option e.
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We 'll re-write the denominator of th ratio as:
cos 2x = (cos x)^2 - (sin x)^2
We'll re-write the difference of squares as a product:
(cos x)^2 - (sin x)^2 = (cos x - sin x)(cos x + sin x)
We'll re-write the function:
f(x) = (sin x-cos x)/(cos x - sin x)(cos x + sin x)
We'll simplify and we'll get:
f(x) = -1/(cos x + sin x)
Now, we'll take limit both sides:
lim f(x) = lim [-1/(cos x + sin x)]
lim f(x) = -1/lim (cos x + sin x)
lim f(x) = -1/(cos pi/4 + sin pi/4)
lim f(x) = -1/(sqrt2/2 + sqrt2/2)
lim f(x) = -1/2sqrt2/2
lim f(x) = -sqrt 2/2
Since the limit of the function is l, then l = -sqrt 2/2, so the good answer is e).
srry, i didnt put x
x-> 45 degrees
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