Question

Find the left hand and right-hand limits of the greatest integer, function $f\left( x \right) = \left[ x \right]$ greatest integer less than or equal to $x$at \[x = k\], where $k$is an integer also, show that $\lim \;f\left( x \right)$ does not exist -

Answer 1

Hint: –This problem is based on limit..a basic property of limit says that if both sides limit gives same result then the limit of the function exist..but if the left and right-hand limit gives different results then the limit of the function not exist..we have to solve the problem by this concept.
 Let us consider the
$
  \lim f\left( x \right) = \lim \left[ x \right] = k \\
  x \to {k^ + }\;\;\;\;\;\;\;\;\;\;\;x \to {k^ + } \\
 $


Complete step by step solution:
Here the given function is \[f\left( x \right) = x\]
So we have to find the left and right-hand limit to solve the problem.
$f\left( x \right) = \left[ x \right]$
$
  \lim f\left( x \right) = \lim \left[ x \right] = k \\
  x \to {k^ + }\;\;\;\;\;\;\;\;\;\;\;x \to {k^ + } \\
 $
$
  \lim f\left( x \right) = \lim \left[ x \right] - k - 1 \\
  x \to {k^ - }\;\;\;\;\;\;\;\;\;\;\;x \to {k^ - } \\
$
$
  \lim f\left( x \right) \ne Lt\;f\left[ x \right] \\
  x \to {k^ + } \\
$
Therefore we can see that both limits do not give the same result so we can say the limit of the function does not exist.
Limit does not exist.


Note: - This type of question to solve first is know the left hand and right-hand limits of greatest integer functions.
Then we have to find the left and right-hand limit of the function..then we can say whether the limit of the function exists or not.