Download App

STD 11 - 12. limits — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 12. limits4 Marks
MCQ
$\mathop {\lim }\limits_{x \to \infty } \,{\left( {\frac{{x + a}}{{x + b}}} \right)^{x + b}} = $
  • A
    $1$
  • B
    ${e^{b - a}}$
  • ✓
    ${e^{a - b}}$
  • D
    ${e^b}$
✓

Answer

Correct option: C.
${e^{a - b}}$
c
(c) $\mathop {{\rm{lim}}}\limits_{x \to \infty } \,{\left( {\frac{{x + a}}{{x + b}}} \right)^{x + b}} = \mathop {{\rm{lim}}}\limits_{x \to \infty } \,{\left( {1 + \frac{{a - b}}{{x + b}}} \right)^{x + b}}$

$ = \mathop {{\rm{lim}}}\limits_{x \to \infty } \,{\left\{ {{{\left( {1 + \frac{{a - b}}{{x + b}}} \right)}^{\frac{{x + b}}{{a - b}}}}} \right\}^{a - b}} = {e^{a - b}}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The differential of ${e^{{x^3}}}$ with respect to $\log x$ is
→
Let $y=y(x)$ be solution of the differential equation $\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y$, with $y(0)=0$. If $y\left(-\frac{2}{3} \log _{e} 2\right)=\alpha \log _{e} 2$, then the value of $\alpha$ is equal to:
→
If $(a, 0); a > 0$ is the point where the curve $y = \sin 2x - \sqrt 3 \, \sin x$ cuts the $x-$ axis first, $A$ is the area bounded by this part of the curve , the origin and the positive $x-$ axis, then
→
Let $\alpha \in(0, \infty)$ and $A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$. If $\operatorname{det}\left(\operatorname{adj}\left(2 \mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right) \cdot \operatorname{adj}\left(\mathrm{A}-2 \mathrm{~A}^{\mathrm{T}}\right)\right)=2^8$, then $(\operatorname{det}(\mathrm{A}))^2$ is equal to :
→
If $A = \left[ {\begin{array}{*{20}{c}}1&1\\0&1\end{array}} \right]$, then ${A^n} = $
→
Consider the equation $x^2+x-n = 0$, where $n \in N$,and $n \in [5,100].$ Then total  Number of different values of $n$, so that the given equation has Integral roots is :-
→
Three numbers form a $G.P.$ If the ${3^{rd}}$ term is decreased by $64$, then the three numbers thus obtained will constitute an $A.P.$ If the second term of this $A.P.$ is decreased by $8$, a $G.P.$ will be formed again, then the numbers will be
→
Let $\mathrm{ABC}$ be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \quad \overrightarrow{B C}=a \hat{i}+b \hat{j}+c \hat{k}$ and $\overrightarrow{\mathrm{AC}}=6 \hat{\mathrm{i}}+\mathrm{d} \hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \mathrm{d}>0$. Then the square of the length of the largest side of the triangle $\mathrm{ABC}$ is....................
→
Solution set of equation

$\left| {1 - {{\log }_{\frac{1}{6}}}x} \right| + \left| {{{\log }_2}x} \right| + 2 = \left| {3 - {{\log }_{\frac{1}{6}}}x + {{\log }_{\frac{1}{2}}}x} \right|$ is $\left[ {\frac{a}{b},a} \right],a,b, \in N,$ then the value of $(a + b)$ is

→
Area of the figure enclosed by $y = {\cos ^{ - 1}}(\cos \,x),x\, \in \,\left[ {2\pi ,4\pi } \right]$ ; $x-$ axis and $y = {\tan ^{ - 1}}\,x + {\tan ^{ - 1}}\,\frac{1}{x}$ is
→