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If we have a limit of the form:

$$\lim_{x \to \infty} \frac{a(x)-b}{c(x)}$$

Is it always possible to write it as $$-\lim_{x \to \infty} \frac{b-a(x)}{c(x)}$$

Can this always be done, or must parts of the expression be/not be functions of $x$?

Edit: the question is motivated by the math in this answer.

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2 Answers 2

up vote 3 down vote accepted

This is an allowed property of limits. Recall

$$\lim_{x \to a} k\cdot f(x) = k\cdot \lim_{x \to a} f(x),$$

So, in your original limit, you can factor out the numerator of the fraction to be: $$a(x)-b=-1\cdot(b-a(x))$$

Now, using Eqn.1, you can simply take out the constant multiple of -1 from the original limit.

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If limits $\lim_{x \to \infty} f(x)$ and $\lim_{x \to \infty} g(x)$ exist, then limit $\lim_{x \to \infty} f(x)g(x)$ also exists.

Take $f(x)=\frac{a(x)-b}{c(x)}$ and $g(x)=-1$.

So, by multiplying the initial limit twice with $-1$, you get the result you want.

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Why bother defining the constant multiple as a function, instead of just simply using the law for a scalar multiple? –  Richard P Feb 25 at 1:31
    
Because that's what I came up with! :) –  frabala Feb 25 at 1:32
    
Okay, just making sure I was not missing something. –  Richard P Feb 25 at 1:32
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