Question:
Given the function:
\[ f(x) = \begin{cases} \frac{(2x^2 - ax +1) - (ax^2 + 3bx + 2)}{x+1}, & \text{if } x \neq -1 \\ k, & \text{if } x = -1 \end{cases} \]
If \( a, b, k \in \mathbb{R} \) and \( f(x) \) is continuous for all \( x \), then the value of \( k \) is:
Given the function:
\[ f(x) = \begin{cases} \frac{(2x^2 - ax +1) - (ax^2 + 3bx + 2)}{x+1}, & \text{if } x \neq -1 \\ k, & \text{if } x = -1 \end{cases} \]
If \( a, b, k \in \mathbb{R} \) and \( f(x) \) is continuous for all \( x \), then the value of \( k \) is:
Show Hint
For continuity at a point \( x = c \), ensure \( \lim_{x \to c} f(x) = f(c) \) by simplifying expressions and canceling terms carefully.
Updated On: Mar 19, 2025
- \( -\frac{1}{3} \)
- \( 6 \)
- \( a - 2 \)
- \( a - 3 \)
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Condition for continuity
For \( f(x) \) to be continuous at \( x = -1 \),
\[
\lim_{x \to -1} f(x) = f(-1).
\]
Substituting \( x = -1 \) in the numerator, simplifying, and equating to \( k \), we get:
\[
k = a - 3.
\]
Was this answer helpful?
0
0
Top Questions on Limits
- Find the equation of the tangent line to the curve \( y = x^2 - 3x + 2 \) at the point \( (1, 0) \).
- The value of \[ \lim_{x \to 1} \frac{x^4 - \sqrt{x}}{\sqrt{x} - 1} \] is:
- \(\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} =\)
- If \( f(x) = \begin{cases} \frac{(e^x - 1) \log(1 + x)}{x^2} & \text{if } x>0 \\ 1 & \text{if } x = 0 \\ \frac{\cos 4x - \cos bx}{\tan^2 x} & \text{if } x<0 \end{cases} \) is continuous at \( x = 0 \), then \(\sqrt{b^2 - a^2} =\)
- Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)
View More Questions
Questions Asked in AP EAMCET exam
- Observe the following reactions:
I. \( \text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g) \)
II. \( \text{Cl}_2(g) \rightarrow 2 \text{Cl}(g) \)
III. \( \text{H}_2\text{O}(l) \rightarrow \text{H}_2O(s) \)
Identify the reactions in which entropy increases.- AP EAMCET - 2024
- Thermodynamics
An inductor and a resistor are connected in series to an AC source of voltage \( 144\sin(100\pi t + \frac{\pi}{2}) \) volts. If the current in the circuit is \( 6\sin(100\pi t + \frac{\pi}{2}) \) amperes, then the resistance of the resistor is:
- AP EAMCET - 2024
- Dual nature of radiation and matter
- If a real valued function \( f: [a, \infty) \to [b, \infty) \) is defined by \( f(x) = 2x^2 - 3x + 5 \) and is a bijection, then find the value of \( 3a + 2b \):
- AP EAMCET - 2024
- Functions
- Let \( \alpha \in \mathbb{R} \). If the line \( (a + 1)x + \alpha y + \alpha = 1 \) passes through a fixed point \( (h, k) \) for all \( a \), then \( h^2 + k^2 = \):
- AP EAMCET - 2024
- general equation of a line
- The length of a metal bar is 20 cm and the area of cross-section is \( 4 \times 10^{-4} \) m\(^2\). If one end of the rod is kept in ice at \( 0^\circ C \) and the other end is kept in steam at \( 100^\circ C \), the mass of ice melted in one minute is 5 g. The thermal conductivity of the metal in Wm\(^{-1}\)K\(^{-1}\) is (Latent heat of fusion = 80 cal/gm)
- AP EAMCET - 2024
- Energy Conservation
View More Questions