Basic maths-calculus-Q20

+1 vote

If \begin{align*} f(x) &= \frac{sin\begin{bmatrix} x \end{bmatrix}}{\begin{bmatrix} x \end{bmatrix}}, \begin{bmatrix} x \end{bmatrix} \ne 0\\ &=0, \begin{bmatrix} x \end{bmatrix} = 0 \end{align*}

Where \begin{bmatrix} x \end{bmatrix} \end{align*} denotes the greatest integer less than or equal to x. Then lim_{x\rightarrow 0} f(x) equals?

(A). 1

(B). 0

(C). -1

(D). None of these

asked May 19 in Basic Maths by gbeditor (44,560 points)
reshown May 24 by gbeditor

1 Answer

+1 vote
 
Best answer

Limit does not exist. 

Right limit : 

i.e. x \rightarrow 0^+

i.e. x = 0+h, h \rightarrow 0

So, [h] = 0, Hence, right limit = 1; (because  \lim_{x \rightarrow 0} sinx/x = 1 )


Left limit : 

i.e. x \rightarrow 0^-

i.e. x = 0-h, h \rightarrow 0

So, [-h] = -1, Hence, left limit = 1; (because  \lim_{x \rightarrow -1} sinx/x = -sin(-1) )

and  sin(-1 \,\,radian) = -0.84147098

Hence, Left limit Not same as right limit, hence, at x = 0, limit does not exist.  

answered May 27 by deepak-gatebook (112,760 points)
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