Question

I need help solving this practice problem from my prep guide

Answer 1

`\lim _(n\to\infty)\text{ }(\frac{3n^5^{}}{6n^6+1})\text{ = 0}`

Step-by-step explanation

Step 1: Given that:

`\sum ^(\infty)_(n\mathop=1)(\frac{3n^5^{}}{6n^6+1})`

Step 2: Expand the limit

`\begin{gathered} \lim _(n\to\infty)(\frac{3n^5^{}}{6n^6+1})\text{ } \\ \text{ = }\lim _(n\to\infty)(n^5)/(n^5)(\frac{3^{}}{6n^{}+(1)/(n^5)}) \\ \text{ = }\lim _(n\to\infty)(\frac{3^{}}{6n^{}+(1)/(n^5)}) \\ =\text{ }\lim _(n\to\infty)(\frac{3^{}}{6(\infty)^{}+(1)/((\infty)^5)}) \\ \text{ = }\lim _(n\to\infty)(\frac{3^{}}{6(\infty)^{}+(1)/(\infty)}) \\ \text{ = }\lim _(n\to\infty)(\frac{3^{}}{6(\infty)^{}+0^{}}) \\ \text{ = }\lim _(n\to\infty)(\frac{3^{}}{6(\infty)^{}^{}})\text{ = 0} \\ \end{gathered}`

Hence,

`\lim _(n\to\infty)((3n^5)/(6n^6+1))\text{ = 0}`