why the limit of this f(x) when x approach infinity is equal to infinity? why the limit of this f(x) when x approach infinity is equal to infinity?:
$$ \lim_{x\to \infty} \frac{x^2 + 4x + 5}{x-1}$$ 
$$ \lim_{x\to \infty} \frac{1 + \frac{4}{x} + \frac{5}{x^2}}{\frac{1}{x}-\frac{1}{x^2}} = \infty  $$ 
I know that all the fraction with X be denominator is equal to Zero but is that mean at the end the $ \lim_{x\to \infty} = \frac{1}{0}$ why it is equal to $ \infty $ ?

$$\lim_{x\to\infty}\left(\frac{x^2+4x+5}{x-1}\right)=$$

The leading term in the denominator of $\frac{x^2+4x+5}{x-1}$ is $x$.
So divide the numerator and denominator by this:

$$\lim_{x\to\infty}\left(\frac{x+4+\frac{5}{x}}{1-\frac{1}{x}}\right)=$$

The expressions $\frac{5}{x}$ and $\frac{1}{x}$ both tend to zero as $x$ approaches $\infty$:

$$\lim_{x\to\infty}\left(\frac{x+4+0}{1-0}\right)=$$
$$\lim_{x\to\infty}\left(\frac{x+4}{1}\right)=$$
$$\lim_{x\to\infty}\left(x+4\right)=$$
$$\lim_{x\to\infty}x=\infty$$
