Calculus

Use the graph to estimate the limit:

lim x->0 sin(3x)/x

When x is in degrees

lim x->0 sin(3x)/x = ________

I thought the the answer was (3*180)/pi but it's not... please help... Thanks

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  1. Use Lim x->0 sin(x)/x =0
    The proof can be done using the sandwich (squeeze) theorem.

    Expand
    sin(3x)
    =3cos²(x) sin(x)-sin³(x)
    =sin(x)(3cos²(x)-1+(1-sin²(x)))
    =sin(x)(3cos²(x)-1+cos²(x))
    =sin(x)(4cos²(x)-1)

    As x->0, cos(x)->1, so the limit becomes
    Lim x->0 3sin(x)/x
    =3 Lim x->0 sin(x)/x
    The latter limit can be quoted as a standard limit, or can be proven using the sandwich theorem.

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  2. The first value I tried was 3... and it didn't work either, I thought that the fact that it said "x is in degrees" affected. It also said at the end: (Give your answer accurate to at least 0.01...

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  3. Sorry, I am used to working with radians without thinking when dealing with limits and calculus. I have not properly read the instructions (would be fatal in exams!).

    To change from degrees to radians, we need to multiply the number of degrees by %pi;/180, so the problem in radians is really:

    Lim x->0 sin(3x*π/180)/x
    =Lim x-> sin(πx/60)/x
    =π/60

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