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Mathematics
14.03.2022 12:31
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Lim ((sin(pi(cosx)))/(x^2)) as x- 0

Lim ((sin(pi(cosx)))/(x^2)) as x->0
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lizzyhearts
lizzyhearts
4,4(10 marks)

Both the numerator and denominator approach 0, so by L'Hopital's rule

\displaystyle\lim_{x\to0}\frac{\sin(\pi\cos x)}{x^2}=\lim_{x\to0}\frac{-\pi\sin x\cos(\pi\cos x)}{2x}

Using the fact that

\displaystyle\lim_{x\to0}\frac{\sin x}x=1

we find

\displaystyle\lim_{x\to0}\frac{-\pi\sin x\cos(\pi\cos x)}{2x}=\lim_{x\to0}\frac{-\pi\cos(\pi\cos x)}2

\pi\cos x\to\pi as x\to0, and \cos\pi=-1, so the limit is π/2.

(I'm wondering if there's a way to do this without L'Hopital's rule, in case you haven't learned it yet...)

ochoamiguel46
ochoamiguel46
5,0(80 marks)

a)

V=p+prt

now we solve for P

V=P (1+rt)

Divide both sides by (1+rt)

P=V÷(1+rt)...answer

b)

P=V÷(1+rt)

P=3,000÷(1+0.06×5)

P=2,307.69

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