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I need to prove these statements, and to learn how to prove it. In my math book, I have a method as $M > 0$ and $x > x_0 \implies f(x) > M$

Suppose we have $y = 3x,\; y = x^2,\; y = kx^2\;\; (k>0)$

Can you teach me how to prove that the limit of those functions when $x$ gets to $\infty$ is $+\infty$?

Guy
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whyguy
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  • invalid question. the limit of the first and third function is NOT infintiy – Guy Mar 03 '14 at 13:45
  • @2012ssohn I misread. I thought it was asking $\frac{y}{x}$ because the direct functions are trivial. And OP, what have you tried? – Guy Mar 03 '14 at 14:01

2 Answers2

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Proof by contradiction. Suppose in any of the cases that f(x) is bounded by some number m, then in all cases you can find an X such that f(x) > m for all x > X.

For example, in first case take X = m.

Tom Collinge
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Let's beging with $f(x)=3x$. We have to prove that: $$\lim_{x \to \infty} f(x) = \infty$$

Given a large number $f(x_0)=M$, with $x_0>0$, for any $x>x_0$ it must follow that $f(x)>M$ (a graph might help here to undestand it). This means that $f$ is not upper bounded.

$$x>x_0$$ $$3x>3x_0$$ $$f(x)>f(x_0)$$ $$f(x)>M$$

You can do the same with the others

Francisco
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