Does the Limit of a Polynomial Diverge as x Approaches Infinity?

[quasar987]

Question: How to show that the limit as x goes to infinity of a given polynomial diverge? We have

$$\lim_{x \rightarrow \infty} a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0$$

but cannot say the this limit is the sum of the limit of each term separetly because none of these limit exist. We cannot do this either

$$\lim_{x \rightarrow \infty} a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0 = \lim_{x \rightarrow \infty} x^n(a_n + \frac{a_{n-1}}{x}+...+\frac{a_1}{x^{n-1}}+\frac{a_0}{x^n})$$


and say the the limit is the product of the limit because one of the limit does not exist. So what then?


Thank you.

 

[arildno]

Let's take the case $$a_{n}>0$$
What you CAN show, using your last equation, is that there exist $$N_{0}$$ so that $$\frac{a_{0}}{x^{n}}\geq{-\frac{a_{n}}{2n}}$$ whenever $$x\geq{N}_{0}$$
Similarly, it exists $$N_{1}$$ so that:
$$\frac{a_{1}}{x^{n-1}}\geq{-\frac{a_{n}}{2n}}$$ whenever $$x\geq{N}_{1}$$
And so on.
Setting N equal to the maximum of these $$N_{i}$$ values, yields the inequality:
$$x^{n}(a_{n}+++\frac{a_{0}}{x^{n}})\geq\frac{x^{n}a_{n}}{2}, x\geq{N}$$

 

[wais]

To show that the limit as x goes to infinity of a given polynomial diverges, we can use the definition of divergence. A polynomial will diverge if the absolute value of its terms increases without bound as x approaches infinity. In other words, the terms of the polynomial become infinitely large as x increases.

We can demonstrate this by looking at the highest degree term in the polynomial, which will dominate the behavior of the polynomial as x approaches infinity. For example, in the polynomial \lim_{x \rightarrow \infty} x^3 + 2x^2 + 3x + 4, the term x^3 will become infinitely large as x increases, causing the entire polynomial to diverge.

Additionally, we can use the limit comparison test to compare the given polynomial to a known divergent function, such as the function f(x) = x, as x approaches infinity. If the limit of the polynomial is equal to the limit of f(x), then the polynomial will also diverge.

In conclusion, to show that the limit of a polynomial diverges, we can use the definition of divergence and compare it to a known divergent function. By doing so, we can see that the terms of the polynomial become infinitely large as x approaches infinity, causing the limit to also become infinitely large and thus diverge.