Discover the Limit of x^2 as x Approaches Infinity | Calculate with Ease

[askor]

What is the ##\lim_{x \to \infty} x^2##?

What I get is:

##\lim_{x \to \infty} x^2##
##= \lim_{x \to \infty} \frac{\frac{1}{x^2}}{\frac{1}{x^2}} x^2##
##= \lim_{x \to \infty} \frac{\frac{x^2}{x^2}}{\frac{1}{x^2}}##
##= \lim_{x \to \infty} \frac{1}{\frac{1}{x^2}}##
##= \frac{1}{\frac{1}{\infty}}##
##= \frac{1}{0}##

 

[PeterDonis]

It looks like you have shown (in a somewhat roundabout way) that the limit is undefined--in other words, that ##x^2## increases without bound as ##x## increases without bound (or, in the more usual sloppy terminology, ##x^2 \rightarrow \infty## as ##x \rightarrow \infty##). Does that seem reasonable to you?

 

[Mark44]

What is the ##\lim_{x \to \infty} x^2##?

What I get is:

##\lim_{x \to \infty} x^2##
##= \lim_{x \to \infty} \frac{\frac{1}{x^2}}{\frac{1}{x^2}} x^2##
##= \lim_{x \to \infty} \frac{\frac{x^2}{x^2}}{\frac{1}{x^2}}##
##= \lim_{x \to \infty} \frac{1}{\frac{1}{x^2}}##

Going from this step to the next, you are saying that ##\lim_{x \to \infty} x^2 = \infty##. In other words you have used a number of unnecessary steps to arrive at pretty much the same thing as you started with.

Also, you should never write either ##\frac 1 {\infty}## or ##\frac 1 0##. In the first, ##\infty## is not a number that can be used in arithmetic expressions, and in the second, ##\frac 1 0## is undefined.

##= \frac{1}{\frac{1}{\infty}}##
##= \frac{1}{0}##

Much more simply, if x grows large without bound, ##x^2## does the same even more rapidly.

 

[fresh_42]

The common ε-δ-definition of limits turns in the case of unlimited sequences to
##∀ r ∈ ℝ ∃ N ∈ ℕ ∀ n ≥ N : x_n > r## for ##\lim_{n→∞} x_n = ∞## (##x_n < r## for ##\lim_{n→∞} x_n = -∞##)