Question:
If
\[
\lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ?
\]
If
\[
\lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ?
\]
Show Hint
When dealing with limits involving rational functions, ensure that the numerator has a factor that cancels with the denominator. Substitute the limiting value into the numerator to find the necessary conditions for cancellation.
Updated On: Apr 7, 2026
- 0
- 5
- -1
- -5
- 1
Show Solution
Verified By Collegedunia
The Correct Option is
Solution and Explanation
We are given the limit:
\[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5 \] Step 1: First, factor the numerator. The expression \( x^2 - ax - b \) should be factorable in the form \( (x - 1) \times (\text{some expression}) \), because the denominator is \( x - 1 \).
We need to ensure that the numerator has a factor of \( (x - 1) \), so substitute \( x = 1 \) into the numerator:
\[ x^2 - ax - b = 1^2 - a(1) - b = 1 - a - b \] For the expression to have a factor of \( (x - 1) \), this must be zero. Therefore:
\[ 1 - a - b = 0 \] \[ a + b = 1 \]
Thus, \( a + b = 1 \).
Therefore, the correct answer is option (E).
\[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5 \] Step 1: First, factor the numerator. The expression \( x^2 - ax - b \) should be factorable in the form \( (x - 1) \times (\text{some expression}) \), because the denominator is \( x - 1 \).
We need to ensure that the numerator has a factor of \( (x - 1) \), so substitute \( x = 1 \) into the numerator:
\[ x^2 - ax - b = 1^2 - a(1) - b = 1 - a - b \] For the expression to have a factor of \( (x - 1) \), this must be zero. Therefore:
\[ 1 - a - b = 0 \] \[ a + b = 1 \]
Thus, \( a + b = 1 \).
Therefore, the correct answer is option (E).
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Magnitude and Directions of a Vector Questions
Let \(a=i+j+2k\) and \(b=i-2j+3k\) be two vectors. Then the unit vector in the direction of \(a-b\) is
- The direction cosines of the vector $\vec{a} = -2\hat{i} + \hat{j} - \hat{k}$ are
- The value of \(λ\) for which the vectors \(i+j-k\)and \(λi+3j+k\) are perpendicular is ?
- Let \(a\) and \(b\) be perpendicular vectors such that \(IaI=√104\) and \(IbI=6\).Then the value of\( Ia-bI\) is ?
- The position vectors of two points P and Q are given by \(OP=2a-b\) and \(OQ=a+3b\) ,respectively.If a point R divides the line joining P and Q internally in the ratio \(1:2\) ,then the position vector of the point R is ?
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions