Question

2B%20%5B%20%5Cdfrac%7B%202%5Csin%282x%29%20%7D%7Bx%7D%20%5D%20%2B%20%5C%3A%20...%20%5C%3A%20%5B%20%5Cdfrac%7B%2010%5Csin%2810x%29%20%7D%7Bx%7D%20%5D%20%5C%5C%20%5C%5C%20Where%20%5C%3A%20%5By%5D%20%5C%3A%20is%20%5C%3A%20the%20%5C%3A%20largest%20%5C%3A%20integer%20%5Cleqslant%20y%20%2C%5C%5C%20Find%20%5C%3A%20the%20%5C%3A%20value%20%5C%3A%20of%20%5C%3A%20%5C%3A%20%5C%3A%20x%5Cxrightarrow%5B%5D%7Blim%7D%200%5C%3A%20%5C%3A%20%5C%3A%20f%28x%29%20%5C%3A%20" id="TexFormula1" title="Let \: \: f(x)= [ \dfrac{ \sin(x) }{x} ] + [ \dfrac{ 2\sin(2x) }{x} ] + \: ... \: [ \dfrac{ 10\sin(10x) }{x} ] \\ \\ Where \: [y] \: is \: the \: largest \: integer \leqslant y ,\\ Find \: the \: value \: of \: \: \: x\xrightarrow[]{lim} 0\: \: \: f(x) \: " alt="Let \: \: f(x)= [ \dfrac{ \sin(x) }{x} ] + [ \dfrac{ 2\sin(2x) }{x} ] + \: ... \: [ \dfrac{ 10\sin(10x) }{x} ] \\ \\ Where \: [y] \: is \: the \: largest \: integer \leqslant y ,\\ Find \: the \: value \: of \: \: \: x\xrightarrow[]{lim} 0\: \: \: f(x) \: " align="absmiddle" class="latex-formula">

Answer 1

Remark
When you take the limit of  $\lim_{ 0} \frac{sin(x)}{x}$ the odd result you get is 1.  Later on you will be able to use calculus to show this. For now just take limits of sin(x)/x and make sure you are feeding radians into your calculator. 

Now the only question is what is this thing doing?
If a is a constant in $\lim_{0 \frac{sin(ax)}{x}$ then the result = a.

So that's basically all you need to know to solve your problem. 

Series
Each term in the series will be
a*(sin(ax)/x) = a * [sin(ax)/x] * 1 = a * a = a^2 

The series will look like this.
1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 There is a way of summing this using n notation, but you could just as easily just add the results.

The formula for this series (if you want a sum) is n*(n+1)*(2n+1) / 6
n = 10
Sum = 10*(11)(21)/6
Sum = 385

Does adding it by hand bring up 385?