Question:
The value of \(\underset{x\rightarrow 3}{lim}\frac{[x]-3}{x-3}\), where \([\cdot ]\) denotes the greatest integer function, is.....
The value of \(\underset{x\rightarrow 3}{lim}\frac{[x]-3}{x-3}\), where \([\cdot ]\) denotes the greatest integer function, is.....
Show Hint
Compare the left-hand and right-hand limits.
Updated On: Oct 1, 2026
- \(\infty\)
- \(1\)
- \(0\)
- does not exist
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Right-hand limit
For \(x\) slightly above \(3\), \([x]=3\), so the fraction is \(\frac{0}{x-3}=0\). The limit is \(0\).
Step 2: Left-hand limit
For \(x\) slightly below \(3\), \([x]=2\), so the fraction is \(\frac{-1}{x-3}\). As \(x\to3^-\), \(x-3\to0^-\), and the fraction tends to \(+\infty\).
Step 3: Compare
The two sided limits differ (\(0\) and \(+\infty\)), so the limit does not exist. Option (D).
Final Answer:
The limit does not exist, option (D). \[ \boxed{\text{(D)}} \]
For \(x\) slightly above \(3\), \([x]=3\), so the fraction is \(\frac{0}{x-3}=0\). The limit is \(0\).
Step 2: Left-hand limit
For \(x\) slightly below \(3\), \([x]=2\), so the fraction is \(\frac{-1}{x-3}\). As \(x\to3^-\), \(x-3\to0^-\), and the fraction tends to \(+\infty\).
Step 3: Compare
The two sided limits differ (\(0\) and \(+\infty\)), so the limit does not exist. Option (D).
Final Answer:
The limit does not exist, option (D). \[ \boxed{\text{(D)}} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Limits Questions
- $ \lim\limits _{x\to 1 } \left(log \,ex\right)^{1/log\,x} $ is equal
- Marks of 5 students of a group are \( 8, 12, 13, 15, 22 \). Find the variance.
- Evaluate \( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \):
The area of the region \( \{(x, y): 0 \leq y \leq x^2 + 1, \, 0 \leq y \leq x + 1, \, 0 \leq x \leq 2\ \) is:}
- The limit of \( \lim_{x \to 0} \frac{\sin x}{x} \) is:
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions