JEE Advanced · Mathematics · 16. Limits
If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then
- A \(a=1, b=4\)
- B \(a=1, b=-4\)
- C \(a=2, b=-3\)
- D \(a=2, b=3\)
Answer & Solution
Correct Answer
(B) \(a=1, b=-4\)
Step-by-step Solution
Detailed explanation
Given: \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\)
\(\begin{array}{l}
\Rightarrow \lim _{x \rightarrow \infty} \frac{x^{2}+x+1-a x^{2}-a x-b x-b}{x+1}=4 \\
\Rightarrow \lim _{x \rightarrow \infty} \frac{(1-a) x^{2}+(1-a-b) x+(1-b)}{x+1}=4
\end{array}\)
For this limit to be finite \(1-a=0 \Rightarrow a=1\) then given limit reduces to
\(\lim _{x \rightarrow \infty} \frac{-b x+(1-b)}{x+1}=4 \Rightarrow \lim _{x \rightarrow \infty} \frac{-b+\frac{(1-b)}{x}}{1+\frac{1}{x}}=4\)
\(\Rightarrow-b=4\) or \(b=-4, \therefore a=1, b=-4\)
\(\begin{array}{l}
\Rightarrow \lim _{x \rightarrow \infty} \frac{x^{2}+x+1-a x^{2}-a x-b x-b}{x+1}=4 \\
\Rightarrow \lim _{x \rightarrow \infty} \frac{(1-a) x^{2}+(1-a-b) x+(1-b)}{x+1}=4
\end{array}\)
For this limit to be finite \(1-a=0 \Rightarrow a=1\) then given limit reduces to
\(\lim _{x \rightarrow \infty} \frac{-b x+(1-b)}{x+1}=4 \Rightarrow \lim _{x \rightarrow \infty} \frac{-b+\frac{(1-b)}{x}}{1+\frac{1}{x}}=4\)
\(\Rightarrow-b=4\) or \(b=-4, \therefore a=1, b=-4\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Mathematics
- Let \(a_{n}\) denote the number of all \(n\)-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let \(b_{n}=\) the number of such \(n\)-digit integers ending with digit 1 and \(c_{n}=\) the number of such \(n\)-digit integers ending with digit 0 .
Question:
Which of the following is correct?JEE Advanced 2012 Hard - The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}\) determines a family of circles withJEE Advanced 2007 Easy
- A computer producing factory has only two plants and . Plant produces and plant produces of the total computers produced. of computers produced in the factory turn out to be defective. It is known that, (computer turns out to be defective given that it is produced in plant ) (computer turns out to be defective given that it is produced in plant ). Where, denotes the probability of an event A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant isJEE Advanced 2016 Medium
- Paragraph:
Read the following passage and answer the questions.
For every function \(f(x)\) which is twice differentiable, these will be good approximation of \(\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}\). Now, if we take \(c=\frac{a+b}{2}\), then using above again, we get \(\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}\) and so on.
We get approximation for value of \(\int_a^b f(x) d x\).Question:
If \(f^{\prime \prime}(x) < 0, \forall x \in(a, b), c(c, f(c))\) is point of maxima where \(c \in(a, b)\), then \(f^{\prime}(c)\) isJEE Advanced 2006 Easy - The option(s) with the value of and that satisfy the following equation is(are)JEE Advanced 2015 Hard
- Let \(y(x)\) be the solution of the differential equation
\(x^2 \frac{d y}{d x}+x y=x^2+y^2, x>\frac{1}{e}\)
satisfying \(y(1)=0\). Then the value of \(2 \frac{(y(e))^2}{y\left(e^2\right)}\) is _____ .JEE Advanced 2025 Easy
More PYQs from JEE Advanced
- Tangents are drawn to the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\), parallel to the straight line \(2 x-y=1\). The points of contact of the tangents on the hyperbola areJEE Advanced 2012 Medium
- Match each coordination compound in List-I with an appropriate pair of characteristics from List-II and select the correct answer using the code given below the lists
\(\left\{\right.\)en \(= H _2 NCH _2 CH _2 NH _2 ;\) atomic numbers \(\left.: T _{ i }=22, Cr =24 ; Co =27 ; Pt =78\right\}\)List – I List – II A. P. Paramagnetic and exhibits ionization isomerism B. Q. Diamagnetic and exhibits cis - trans isomerism C. R. Paramagnetic and exhibits cis - trans isomerism D. S. Diamagnetic and exhibits ionization isomerism JEE Advanced 2014 Medium - of is combusted in a fixed volume bomb calorimeter with excess of at and into . During the reaction, temperature increases from to . If heat capacity of the bomb calorimeter and enthalpy of formation of are and at , respectively, the calculated standard molar enthalpy of formation of at is . The value of is___[Given: Gas constant ]JEE Advanced 2022 Hard
- Two coherent monochromatic point sources and of wavelength are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is . Which of the following options is/are correct?
JEE Advanced 2017 Medium - Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter \(D\) of a tube. The measured value of \(D\) is:
JEE Advanced 2025 Easy - A charge is surrounded by a closed surface consisting of an inverted cone of height and base radius , and a hemisphere of radius as shown in the figure. The electric flux through the conical surface is (in units). The value of is _________.
JEE Advanced 2022 Easy