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  • Try this! - Limit , continuity and differentiability - JEE Main-3

\lim_{x\rightarrow 0}\frac{log\, x^{n}-\left [ x \right ]}{\left [ x \right ]},n\in N,([x] denotes\; \, greatest\, \; integer\, \; less \, \; than\, \; or\, \; equal\, \; to\, \; x) 

  • Option 1)

    has value – 1

  • Option 2)

    has value 0

  • Option 3)

    has value 1

  • Option 4)

    does not exist

 

Answers (1)

As we learnt in 

Condition for discontinuity -

1. \:L\neq R

\lim_{x\rightarrow a^{-}}\:f(x)=\lim_{x\rightarrow a^{+}}\:f(x)

limit of function at x = a does not exist.

2.\:L=R\neq V

limit exist but not equal to  x = a

-

 

\lim_{n \to 0} \frac{log x^{n}-[x]}{[x]}

\lim_{n \to 0} \frac{nlog x-[x]}{[x]}

Since log x defined for x > 0 and for x\rightarrow 0^{+} [x] = 0 

and x\rightarrow 0^{-}   [x]= - 1

so LHL \neq RHL 


Option 1)

has value – 1

this is incorrect option

Option 2)

has value 0

this is incorrect option

Option 3)

has value 1

this is incorrect option

Option 4)

does not exist

this is correct option

Posted by

Vakul

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