AP EAMCET · Maths · Limits
If \(a>0 .[\cdot]\) denotes greatest integer function and \(\lim _{x \rightarrow a^{-}}\left(\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\right)=k, \lim _{x \rightarrow a^{+}}\left(\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\right)=l\), then
- A \(k=l\)
- B \(k-l=1\)
- C \(l-k=1\)
- D \(l=a^2, k\) does not exist
Answer & Solution
Correct Answer
(B) \(k-l=1\)
Step-by-step Solution
Detailed explanation
For \(x \rightarrow a^{-}\) \(\frac{x}{a} 1 \Rightarrow\left[\frac{x}{a}\right]=1\) \(\lim _{x \rightarrow a^{+}} \frac{|x|^3}{a}=\frac{a^3}{a}=a^2\) \(\begin{aligned} & k=a^2-0=a^2 \\ & l=\left(a^2-1\right) \\ & k-l=a^2-\left(a^2-1\right) \\ & =1\end{aligned}\) Note: In…
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