Question #60777

Question

1024 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-15T21:02:57

$lim_(xrarroo)(10x^2+7x-3)/sqrt(5x^4+2x^2-5x)=2sqrt5$

Factor the largest degree of $x$ from the numerator and denominator.

$=lim_(xrarroo)(x^2(10+7/x-3/x^2))/sqrt(x^4(5+2/x^2-5/x^3))$

Pulling out $sqrt(x^4)$ as $x^2$ :

$=lim_(xrarroo)(x^2(10+7/x-3/x^2))/(x^2sqrt(5+2/x^2-5/x^3))$

These cancel:

$=lim_(xrarroo)(10+7/x-3/x^2)/(sqrt(5+2/x^2-5/x^3))$

As $x$ goes to infinity, any term like $3/x$ or $-5/x^3$ will approach $0$ :

$=(10+7/oo-3/oo)/sqrt(5+2/oo-5/oo)$

$=(10+0+0)/sqrt(5+0+0)$

$=10/sqrt5$

$=(10sqrt5)/5$

$=2sqrt5$