Let f : (0, ) (2,20) be twice differentiable function such that _… - Vidyadip

MCQ

Let $f : (0, \infty) \to (2,20)$ be twice differentiable function such that $\mathop {\lim }\limits_{x \to \infty } (f(x) + f'(x) + f^{''}(x)) = \mathop {\lim }\limits_{x \to \infty } g(x)$

where $\mathop {\lim }\limits_{x \to \infty } g(x)$ exists and equal to $5$, then $\mathop {\lim }\limits_{x \to \infty } (f(x) - g(x))$ equal to

A
$5$
B
$7$
✓
$0$
D
Does not exists

✓

Answer

Correct option: C.

$0$

c
Given $\mathop {\lim }\limits_{x \to \infty } \left( {f(x) + {f^\prime }(x) + {f^{\prime \prime }}(x)} \right) = 5$

$\therefore \mathop {\lim }\limits_{x \to \infty } f(x) = \mathop {\lim }\limits_{x \to \infty } \frac{{f(x) \cdot {e^x}}}{{{e^x}}}$

$ = \mathop {\lim }\limits_{x \to \infty } \frac{{\left( {f(x) + {f^\prime }(x)} \right){e^x}}}{{{e^x}}}$

(By L'Hópital rule)

$ = \mathop {\lim }\limits_{x \to \infty } \left( {f(x) + \frac{{{f^\prime }(x){e^x}}}{{{e^x}}}} \right)$

$ = \mathop {\lim }\limits_{x \to \infty } \left( {f(x) + \frac{{\left( {{f^\prime }(x) + {f^{\prime \prime }}(x)} \right){e^x}}}{{{e^x}}}} \right)$

$ = \mathop {\lim }\limits_{x \to \infty } \left( {f(x) + {f^\prime }(x) + {f^\prime }(x)} \right) = 5$

$ \Rightarrow \mathop {\lim }\limits_{x \to \infty } (f(x) - g(x)) = 0$

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

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A parallelopiped is formed by planes drawn through the point (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of a diagonal of the parallelopiped is:

$7$
$\sqrt{38}$
$\sqrt{155}$
$\text{none of these}$

Let $A\,=\,\{\,x\,\in \,R\,:\,x$ is not a positive int eger $\}$ Define a function $f\,:\,A\,\to \,R$ as $f\,(x)\, = \frac{{2x}}{{x - 1}}$ then $f$ is

If $A = \left[ {\begin{array}{*{20}{c}}\alpha &0\\1&1\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}1&0\\5&1\end{array}} \right]$, then value of $\alpha $for which ${A^2} = B$, is

Area bounded by the curve $xy - 3x - 2y - 10 = 0,$ $x -$ axis and the lines $x = 3,x = 4$ is

Let $f(x)=\left\{\begin{array}{cl}x^2 \sin \left(\frac{1}{x}\right) & , x \neq 0 \\ 0 & , x=0\end{array} ;\right.$ Then at $x=0$

A man tosses a coin $10$ times, scoring $1$ point for each head and $2$ points for each tail. Let $P(K)$ be the probability of scoring at least $K$ points. The largest value of $K$ such that $P(K) > \frac{1}{2}$ is

Let $f: R \rightarrow R$ satisfy $f(x+y)=2^{x} f(y)+4^{y} f(x), \forall x$, $y \in R$. If $f(2)=3$, then $14 \cdot \frac{f^{\prime}(4)}{f^{\prime}(2)}$ is equal to

lf Rolle's theorem holds for the function $f(x) =2x^3 + bx^2 + cx, x \in [-1, 1],$ at the point $x = \frac {1}{2},$ then $2b+ c$ equals

Minimum distance between parabola $y^2 = 8x$ and its image with respect to line $x + y + 4 = 0$ is-

If ${x^3} + 8xy + {y^3} = 64$, then ${{dy} \over {dx}} = $