Question

Evaluate lim

x→4

√2

x+

1−

3

x

2−

x−

12​

Answer 1

functions f and g are differentiable on an open interval I containing a. Assume also that g′(x)≠0 in I if x≠a. Then

limx→a

f(x)

g(x)

= limx→a

f′(x)

g′(x)

so long as the limit is finite, +∞, or −∞. Similar results hold for x→∞ and x→−∞.

THEOREM 2 (l'Hopital's Rule for infinity over infinity): Assume that functions f and g are differentiable for all x larger than some fixed number. If limx→a f(x)=∞ and limx→a g(x)=∞, then

limx→a

f(x)

g(x)

= limx→a

f′(x)

g′(x)

so long as the limit is finite, +∞, or −∞. Similar results hold for x→∞ and x→−∞