I can think of a number of ways to prove that limn→∞(4n−n4)=∞

Question

Pubephenedsjq · Accepted Answer

The second term is very small compared to the first as n gets large. Specifically

lim_{n \to \infty} 4^{n} - n^{4} 4^{n} = lim_{n \to \infty} (1 - n^{4} 4^{n})

= 1 - lim_{n \to \infty} \frac{n^{4}}{4^{n}}

The limit on the right can be shown to be zero in several ways, such as applying L'hopital's rule several times. So the ratio goes to 1 as n goes to infinity. As a result,

lim_{n \to \infty} 4^{n} - n^{4} = lim_{n \to \infty} {4^{n} - n^{4} o v e r 4^{n}} 4^{n}

= lim_{n \to \infty} {4^{n} - n^{4} 4^{n}} lim_{n \to \infty} 4^{n}

= \infty

Alternatively, the above limit shows that for n large enough you have

{4^{n} - n^{4} 4^{n}} > \frac{1}{2}

So since

lim_{n \to \infty} 4^{n} = \infty, lim_{n \to \infty} 4^{n} - n^{4} = \infty

as well

I can think of a number of ways

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New Questions in Calculus 1