tmt said:I have this limit: $$\lim_{n\to\infty} (-1)^n\,n^3 + 2^{-n}$$
and I'm unsure how to evaluate it or how to apply L'hopital's rule to this limit.
The limit does not exist as x approaches infinity because the expression oscillates between positive and negative values infinitely.
To evaluate the limit, we can split the expression into two separate limits, one for the first term and one for the second term. We can then use the limit laws to simplify each term and evaluate them separately.
Yes, the value of n does affect the limit. As n increases, the first term of the expression, (-1)^n(n^3), will also increase, while the second term, 2^-n, will decrease. This causes the expression to oscillate between positive and negative values, making the limit undefined.
This is a divergent limit, as the expression does not approach a single value as x approaches infinity. Instead, it oscillates between positive and negative values infinitely.
No, L'Hopital's rule cannot be applied to this limit because it is not in an indeterminate form. Both the numerator and denominator approach infinity, but at different rates, making it impossible to apply L'Hopital's rule.