Limit definition when x tends to infinity

Question

Using the limit definition, show that:

$\lim\limits_{x\to\infty} \frac{x+7}{3x^2+2}=0$

I get blocked when I use the equation:

Formal definition of limit

if I apply the equation would look like this:

| x - ∞ | < ð → | $\frac{x+7}{3x^2+2}-0 < ε)$

Answer 1

The definition you give is for $x$ tending to $x_0$. The definition when $x$ tends to $\infty$ is:

Definition $f(x)$ tends to $L$ as $x$ tends to $ \infty$ if and only if $$ \forall \epsilon >0, \exists \delta>0 \ \mbox{ such that} \ \forall x \ \mbox{where} \ x>\delta, \\ |f(x)-L|<\epsilon$$

Can you use this to solve the question?