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  • Evaluate given limit: <img alt="\lim _{x \rightarrow \infty} \frac{6 x^4+56 x^3+29 x^2}{2 x^3+102 x^2+54 x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%20%5

Evaluate given limit: \lim _{x \rightarrow \infty} \frac{6 x^4+56 x^3+29 x^2}{2 x^3+102 x^2+54 x}

Option: 1

0


Option: 2

1


Option: 3

\infty


Option: 4

Limit does not exist.


Answers (1)

best_answer

\lim _{x \rightarrow \infty} \frac{6 x^4+56 x^3+29 x^2}{2 x^3+102 x^2+54 x}

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}

\text { Now, } a=4 \text { and } \mathrm{b}=3 \text {, which means } \mathrm{a}>\mathrm{b} \text {, hence according to result: }

 

\lim _{x \rightarrow \infty} \frac{6 x^4+56 x^3+29 x^2}{2 x^3+102 x^2+54 x}=\infty

Posted by

Ritika Jonwal

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