# Infinity Question

Why does $$\displaystyle \lim _{ x\rightarrow \infty }{ \frac { 8 }{ x } } =0$$ but $$\displaystyle \lim _{ x\rightarrow \infty }{ x\cdot 0 } =8$$ is not true? What kind of number is zero?

Note by Charles White
2 years, 1 month ago

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The $$2^{\text{nd}}$$ limit isn't a valid one,the function you're taking the limit of is $$0$$ for all $$x$$ (anything multiplied by $$0$$ will give $$0$$),but the first one is valid since the function $$\frac{8}{x}$$ approaches $$0$$ as $$x\rightarrow\infty$$.I think you multiplied both sides by $$x$$ then switched the limit to the $$\text{RHS}$$,this is not valid as you can't multiply by the variable in a limit equation as you did,multiplying by a constant is valid though.

- 2 years, 1 month ago

One of the problems if we went ahead and accepted the second limit to be true is that we can replace 8 with another number and still get an equivalent answer.

So, the second limit would be simultaneously equal to two numbers - which creates a contradiction, which is not acceptable in mathematics.

- 2 years, 1 month ago

I think 0 is not a number. It shows there is nothing.Just like infinite which shows there is everything. But other numbers show something.

- 2 years, 1 month ago

I think you can't do this because the limit is written

$$\lim _{ x \rightarrow x_0} {f(x)}=l$$

and so 8 and x are strictly linked (they are f(x)) in your equation and you can't do what you have done.

- 2 years, 1 month ago