TS EAMCET · Maths · Limits

$\lim _{x \rightarrow \infty}\left[\frac{x^2+x+3}{x^2-x+2}\right]^x$ is equal to

A $\infty$
B $e$
C $e^4$
D $e^2$

Verified Solution

Answer & Solution

Correct Answer

(D) $e^2$

Step-by-step Solution

Detailed explanation

Given,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

If $\frac{x^5-5}{x^3+x^2}=f(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$, then the larger value of $K$ for which $f(K)+A+B+C=1$, is

TS EAMCET 2020 Medium
$\int \frac{d x}{\left(x^2-a^2\right)^{\frac{3}{2}}}$ is equal to

TS EAMCET 2021 Medium
If $A + B + C = 270 °,$ then $\cos 2 A + \cos 2 B + \cos 2 C + 4 \sin A \sin B \sin C =$

TS EAMCET 2019 Medium
Let $f: R \rightarrow R$ be defined by $f(x)=\left\{\begin{array}{ccc}\alpha+\frac{\sin [x]}{x}, & \text { if } & x>0 \ 2, & \text { if } & x=0 \ \beta+\left[\frac{\sin x-x}{x^3}\right], & \text { if } & x < 0\end{array}\right.$ where, $[x]$ denotes the integral part of $x$. If $f$ continuous at $x=0$, then $\beta-\alpha$ is equal to

TS EAMCET 2012 Medium
$\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)$ \[ \left(1+\cos \frac{7 \pi}{8}\right)= \]

TS EAMCET 2019 Easy
For the parabola $y=x^2-3 x+2$, match the items in list-1 to that of the items in list-2. S is a focus, Z is intersection of axis and directrix, P is one end point of latus rectum, Q is the point on the parabola at which tangent is parallel to X -axis
List-1 List-2
A P I $(2,0)$
B Q II $\left(\frac{3}{2},-\frac{1}{4}\right)$
C S III $\left(\frac{3}{2}, 0\right)$
D Z IV $\left(\frac{3}{2},-\frac{1}{2}\right)$
V $\left(0, \frac{3}{2}\right)$

TS EAMCET 2025 Medium

	List-1		List-2
A	P	I	\((2,0)\)
B	Q	II	\(\left(\frac{3}{2},-\frac{1}{4}\right)\)
C	S	III	\(\left(\frac{3}{2}, 0\right)\)
D	Z	IV	\(\left(\frac{3}{2},-\frac{1}{2}\right)\)
		V	\(\left(0, \frac{3}{2}\right)\)

Explore more questions on app

From TS EAMCET

A plane electromagnetic wave of electric and magnetic fields $E_0$ and $B_0$ respectively incidents on a surface. If the total energy transferred to the surface in a time of ' $t$ ' is ' $U$, then the magnitude of the total momentum delivered to the surface for complete absorption is

TS EAMCET 2024 Easy
The random variable takes the values $1,2,3$, $\ldots, m$. If $P(X=n)=\frac{1}{m}$ to each $n$, then the variance of $X$ is

TS EAMCET 2013 Hard
The general solution of the differential equation $\frac{d y}{d x}=1+x+y+x y$ is

TS EAMCET 2018 Easy
If all the vertices of an equilateral triangle lie on the parabola $y^2=16 x$ and one of them coincides with the vertex of that parabola, then the length of the side of that triangle is

TS EAMCET 2020 Medium
Metamerism can be exhibited by the compounds containing

TS EAMCET 2021 Easy
Eight spherical rain drops of the same mass and radius are falling down with a terminal speed of $6 \mathrm{~cm}-\mathrm{s}^{-1}$. If they coalesce to form one big drop, what will be the terminal speed of bigger drop? (Neglect the buoyancy of the air)

TS EAMCET 2009 Easy

Explore more questions on app

\(\lim _{x \rightarrow \infty}\left[\frac{x^2+x+3}{x^2-x+2}\right]^x\) is equal to

Answer & Solution

Step-by-step Solution

See the Complete Solution