Difficulty finding these two limits

dimanche 1 septembre 2013

1. The problem statement, all variables and given/known data

Given that

lim f(x) = 4

x → 2

lim g(x) = -2

x → 2

lim h(x) = 0

x → 2

find the limits that exist for the problems below. If the limits do not exist, explain why.

c)

lim √f(x)

x → 2

e)

lim (f(x)/g(x))

x → 2

2. Relevant equations

lim √f(x)

x → a

=

√(lim f(x))

x → a

and

lim (f(x)/g(x))

x → a

=

(lim f(x))

x → a

/

(lim g(x))

x → a

3. The attempt at a solution

c) Since

lim √f(x)

x → 2

= 4,

√(lim f(x))

x → 2

= √4 = +2 and -2

Therefore, since 2 ≠ -2, the limit does not exist, since I don't think there can be two limits for one number.

e) Since

lim (f(x)/g(x))

x → a

=

(lim f(x))

x → a

/

(lim g(x));

x → a

lim (g(x)/h(x))

x → 2

=

(lim g(x))

x → 2

/

(lim h(x));

x → 2

= -2/0 = Undefined or ∞

The limit might equal +∞ or -∞ here, but since I don't actually have the function or the graph of the function to determine if the limit is +∞ or -∞, I will say the limit does not exist because division by 0 is not defined.