1. ## evaluate limits

Hi all! Here is my question which I am stuck on:
(its actually in 2 parts)
(a) If $\displaystyle f\rightarrow\mathbb{R}$, with D= { x | x is irrational} and $\displaystyle f(x) =\frac{2x}{x-3}$, evaluate $\displaystyle \lim_{x\to\infty}f(x)$ and prove the result.

(b) If $\displaystyle f:\mathbb{Q}\rightarrow\mathbb{R}$ is defined by $\displaystyle f(x)=\frac{x^2+1}{x-2}$, evaluate $\displaystyle \lim_{x\to-\infty}f(x)$ and prove the result.

What do I do?
Thanks

2. Originally Posted by binkypoo
Hi all! Here is my question which I am stuck on:
(its actually in 2 parts)
(a) If $\displaystyle f\rightarrow\mathbb{R}$, with D= { x | x is irrational} and $\displaystyle f(x) =\frac{2x}{x-3}$, evaluate $\displaystyle \lim_{x\to\infty}f(x)$ and prove the result.

(b) If $\displaystyle f:\mathbb{Q}\rightarrow\mathbb{R}$ is defined by $\displaystyle f(x)=\frac{x^2+1}{x-2}$, evaluate $\displaystyle \lim_{x\to-\infty}f(x)$ and prove the result.

What do I do?
Thanks

Here's something to get you started on the first one:

$\displaystyle \frac{2x}{x - 3} = \frac{x(2)}{x(1 - \frac{3}{x})} = \frac{2}{1 - \frac{3}{x}}$

3. I see that but I dont know where to go from there, ie using the definition of limits to infinity to prove it !

4. Originally Posted by binkypoo
I see that but I dont know where to go from there, ie using the definition of limits to infinity to prove it !
What happens to $\displaystyle \frac{3}{x}$ as $\displaystyle x \to \infty$?

Using this, what happens to

$\displaystyle \frac{2}{1 - \frac{3}{x}}$ as $\displaystyle x \to \infty$?

5. Originally Posted by binkypoo
Hi all! Here is my question which I am stuck on:
(its actually in 2 parts)
(a) If $\displaystyle f\rightarrow\mathbb{R}$, with D= { x | x is irrational} and $\displaystyle f(x) =\frac{2x}{x-3}$, evaluate $\displaystyle \lim_{x\to\infty}f(x)$ and prove the result.

(b) If $\displaystyle f:\mathbb{Q}\rightarrow\mathbb{R}$ is defined by $\displaystyle f(x)=\frac{x^2+1}{x-2}$, evaluate $\displaystyle \lim_{x\to-\infty}f(x)$ and prove the result.

What do I do?
Thanks
b)
$\displaystyle \frac{x^2 + 1}{x - 2} = \frac{x^2 - 4 + 5}{x - 2}$

$\displaystyle = \frac{x^2 - 4}{x - 2} + \frac{5}{x - 2}$

$\displaystyle = \frac{(x + 2)(x - 2)}{x - 2} + \frac{5}{x - 2}$

$\displaystyle = x + 2 + \frac{5}{x - 2}$.

Now take the limit as $\displaystyle x \to -\infty$.

6. so because $\displaystyle \lim_{x\to -\infty}x = -\infty$, it doesnt really matter what the limit of the other two terms are, it will still be $\displaystyle -\infty$ right?

7. Well, it would matter if the limit of the other terms went to $\displaystyle +\infty$ (then you would have to be more careful), but since it obviously doesn't, no, it doesn't matter.