Finding the limit of the following sequence: $\left(\frac{2m^2 + m + 4}{2m^2 + 3m + 5}\right)^{3m-3}$

Using the correct hint of the user @gt6989b, I add my solution to your limit.

$$\left(\frac{2m^2 + m + 4}{2m^2 + 3m +5}\right)^{3m-3}=\left[\left(1+\left(\frac{1}{1-\dfrac{2m^2+3m+5}{2m+1}}\right)\right)^{\left(\dfrac{1-\dfrac{2m^2+3m+5}{2m+1}}{1-\dfrac{2m^2+3m+5}{2m+1}}\right)}\right]^{3m-3}$$

$$\left[\left(1+\left(\frac{1}{1-\dfrac{2m^2+3m+5}{2m+1}}\right)\right)^{\left(\dfrac{1}{1-\dfrac{2m^2+3m+5}{2m+1}}\right)}\right]^{\dfrac{6m^2+\ldots}{-2m^2+\cdots}}$$

Hence we have $$\lim_{m \to \infty} \left(\frac{2m^2 + m + 4}{2m^2 + 3m +5}\right)^{3m-3}=e^{-3}=1/e^3$$


HINT

$$ \frac{2m^2 + m + 4}{2m^2 + 3m +5} = 1 - \frac{2m + 1}{2m^2 + 3m +5} $$

Tags:

Limits

Calculus

Sequences And Series

Related

Proof of $\mathbb E[\exp( \lambda XY)]=\mathbb E[\exp( \lambda^2 X^2)/2]$, where $X,Y$ are independent standard normal random variables Is this alternative representation of $f(x)=xe^x$ as Maclaurin series correct? $a+b+c=x+y+z$ and $abc=xyz$ , in which each two of them unequal. Can a curve fill a disk? Convergence of positive sequence in which each term is less than the average of preceding 2 terms When $A$ is $3\times3$, prove that $A^2=0$ iff $A$ has rank less than or equal one and trace zero Term for the sum of products of a number's prime factors, excluding one factor per product. Negation of a Basic Logic Statement Is this a trivial counterexample to basic Rosen number theory exercise? How to prove that when $n$ is even, and $A = (a_{ij})_{i,j}$, $1\leq i,j\leq n$ to show that $\mathrm{det} A = 1$? If $\alpha=\sqrt[3]{2}$ and $p,q,r\in\mathbb{Q}$ then show $p+q\alpha+r\alpha^2$ is a subfield of $\mathbb{C}$ Translate this logical statement into natural language.

Recent Posts