If  f(x)=[x]-\left [ \frac{x}{4}\right ],x\:\:\varepsilon \:\:Rwhere  [x]  denotes   the greatest integer function , then :

  • Option 1)

    f is continuous at x=4.

  • Option 2)

    \lim_{x\rightarrow 4+}f(x)  exist but    \lim_{x\rightarrow 4-}f(x) does not exist.

  • Option 3)

    Both  \lim_{x\rightarrow 4-}f(x)  and \lim_{x\rightarrow 4+}f(x)   exist but are not equal .

  • Option 4)

      \lim_{x\rightarrow 4-}f(x)  exist but \lim_{x\rightarrow 4+}f(x)  does not exist .

 

Answers (1)

f(x)=[x]-\left [ \frac{x}{4}\right ],x\:\:\varepsilon \:\:R

\\ \lim_{h\rightarrow o^{+}}\left [ 4+h \right ]-\left [ \frac{4+h}{4} \right ]\\\\\:=4-1=3

\\ \lim_{h\rightarrow o^{-}}\left [ 4-h \right ]-\left [ \frac{4-h}{4} \right ]\\\\\:=3-0=3

\lim_{h\rightarrow o^{+}}=\lim_{h\rightarrow o^{-}}

hence f(x)  is continous at  x=4

 


Option 1)

f is continuous at x=4.

Option 2)

\lim_{x\rightarrow 4+}f(x)  exist but    \lim_{x\rightarrow 4-}f(x) does not exist.

Option 3)

Both  \lim_{x\rightarrow 4-}f(x)  and \lim_{x\rightarrow 4+}f(x)   exist but are not equal .

Option 4)

  \lim_{x\rightarrow 4-}f(x)  exist but \lim_{x\rightarrow 4+}f(x)  does not exist .

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