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  • Evaluate given limit: <img alt="\lim _{x \rightarrow \infty} \frac{63 x^3+57 x^2+82 x}{26 x^4+74 x^3+53 x^2}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%20%

Evaluate given limit: \lim _{x \rightarrow \infty} \frac{63 x^3+57 x^2+82 x}{26 x^4+74 x^3+53 x^2}

Option: 1

\frac{63}{26}


Option: 2

0


Option: 3

1


Option: 4

Limit does not exist.


Answers (1)

best_answer

\lim _{x \rightarrow \infty} \frac{63 x^3+57 x^2+82 x}{26 x^4+74 x^3+53 x^2}
For solving this equation, a common result can be used:
If a, b are positive integers and m_0, n_0 \neq 0 are non-zero real numbers, then
$$ \lim _{x \rightarrow \infty} \frac{m_0 x^a+m_1 x^{a-1}+\ldots+m_{a-1} x+m_a}{n_0 x^b+n_1 x^{b-1}+\ldots+n_{b-1} x+n_b}= \begin{cases}0 & \text { if } a<b \\ \frac{m_0}{n_0} & \text { if } a=b \\ \infty & \text { if } a>b \text { when } m_0 n_0>0 \\ -\infty & \text { if } a>b \text { when } m_0 n_0<0\end{cases}
Now, \mathrm{a}=3$ and $\mathrm{b}=4, which means \mathrm{a}<\mathrm{b}, hence according to result:
\lim _{x \rightarrow \infty} \frac{63 x^3+57 x^2+82 x}{26 x^4+74 x^3+53 x^2}=0

Posted by

Deependra Verma

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