If f(x)=[x]- [frac x4 ] , xin R , where [x] denotes the greatest integer function, then :

Limits & Continuity: If f(x)=[x]- [frac x4 ] , xin R , where [x] denotes the greatest… | JEE Main 2019

JEE Main 2019 (Online) 9th April Evening ShiftQuestion 1

If $f(x)=[x]-\left[\frac x4\right]$, $x\in R$, where $[x]$ denotes the greatest integer function, then :

Limits & Continuity

$\lim_{x\to4+}f(x)$ exists but $\lim_{x\to4-}f(x)$ does not exist$f$ is continuous at $x=4$✓ Correct answer$\lim_{x\to4-}f(x)$ exists but $\lim_{x\to4+}f(x)$ does not existBoth $\lim_{x\to4-}f(x)$ and $\lim_{x\to4+}f(x)$ exist but are not equal

Solution

Concept. Both $[x]$ and $\big[\frac x4\big]$ jump by $1$ at $x=4$, so the jumps cancel in the difference. Formulas used. (1) $[t]=m$ for $m\le t<m+1$Step 1 - Left limit (1). For $3<x<4$: $[x]=3$, $\frac x4\in(\frac34,1)$ so $\big[\frac x4\big]=0$. $f\to3$. Step 2 - Right limit. For $4<x<5$: $[x]=4$, $\frac x4\in(1,\frac54)$ so $\big[\frac x4\big]=1$. $f\to3$. Step 3 - Value. $f(4)=4-1=3$. Step 4 - Conclude. Left $=$ right $=f(4)=3$: $f$ is continuous at $4$. Answer: B

Limits & Continuity - JEE Mathematics question (JEE Main 2019 (Online) 9th April Evening Shift)

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