If [cdot] denotes the greatest integer functi | vedclass

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Question

(A) We are given the limit: $\lim _{x \rightarrow \frac{-3}{5}} \frac{1}{x}\left[\frac{-1}{x}\right]$.As $x \rightarrow \frac{-3}{5}$,the term $\frac{

Vedclass · Accepted Answer

$\frac{-5}{3}$

Vedclass · Answer

(A) We are given the limit: $\lim _{x ightarrow \frac{-3}{5}} \frac{1}{x}\left[\frac{-1}{x}ight]$.As $x ightarrow \frac{-3}{5}$,the term $\frac{-1}{x} ightarrow \frac{-1}{-3/5} = \frac{5}{3}$.Since $\frac{5}{3} = 1.66...$,it lies in the interval $[1, 2)$.By the definition of the greatest integer function,$[\frac{5}{3}] = 1$.Substituting these values into the expression,we get:$\lim _{x}$ ${ightarrow \frac{-3}{5}} \frac{1}{x}\left[\frac{-1}{x}ight] = \left(\frac{1}{-3/5}ight) 	imes [\frac{5}{3}] = \frac{-5}{3} 	imes 1 = \frac{-5}{3}$.

If $[\cdot]$ denotes the greatest integer function,then $\lim _{x \rightarrow \frac{-3}{5}} \frac{1}{x}\left[\frac{-1}{x}\right]=$

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