Question

Limit as x approaches infinity: 2x/(3x²+5)

Answer 1

$\bf \lim\limits_{x\to \infty}~\cfrac{2x}{3x^2+5}\implies \cfrac{\lim\limits_{x\to \infty}~2x}{\lim\limits_{x\to \infty}~3x^2+5}$

now, by traditional method, as "x" progresses towards the positive infinitity, it becomes 100, 10000, 10000000, 1000000000 and so on, and notice, the limit of the numerator becomes large.

BUT, notice the denominator, for the same values of "x", the denominator becomes larg"er" than the numerator on every iteration, ever becoming larger and larger, and yielding a fraction whose denominator is larger than the numerator.

as the denominator increases faster, since as the lingo goes, "reaches the limit faster than the numerator", the fraction becomes ever smaller an smaller ever going towards 0.

now, we could just use L'Hopital rule to check on that.

$\bf \lim\limits_{x\to \infty}~\cfrac{2x}{3x^2+5}\stackrel{LH}{\implies }\lim\limits_{x\to \infty}~\cfrac{2}{6x}$

notice those derivatives atop and bottom, the top is static, whilst the bottom is racing away to infinity, ever going towards 0.