Given that lim x → 3 f ( x ) = 4, lim x → 3 g ( x ) = − 2, lim x → 3 h ( x ) = 0, find the limits, if they exist

Question

Given that

 limx→3f(x)=4        limx→3g(x)=−2        limx→3h(x)=0,

find the limits, if they exist. (If an answer does not exist, enter DNE.)

(a) limx→3 [f(x)+3g(x)] 

(b) limx→3 [g(x)]^3 

(c) limx→3 √f(x) 

(d) limx→3 3f(x)g(x) 

(e) limx→3 g(x)h(x) 

(f) limx→3 g(x)h(x)f(x) 

Answer 1

Given that lim_{x → 3} f ( x ) = 4, lim_{x → 3} g ( x ) = −2, lim_{x → 3} h ( x ) = 0, find the limits, if they exist. (If an answer does not exist, enter DNE.). (a) lim_x→3 [f(x) + 3g(x)]; (b) lim_x→3 [g(x)]³; (c) lim_x→3 √f(x); (d) lim_x→3 3f(x)g(x); (e) lim_x→3 g(x)h(x); (f) lim_x→3 g(x)h(x)f(x)

Detailed Solutions:

(a) lim_x→3 [f(x) + 3g(x)] = lim_x→3 f(x) + 3 lim_x→3 g(x) = 4 + 3 (−2) = 4 − 6 = −2

(b) lim_x→3 [g(x)]³ = ( lim_x→3 g(x) )³ = (−2)³ = −8

(c) lim_x→3 √f(x) = √ (lim_x→3f(x)) = √ (4) = = √4 = 2

(d) lim_x→3 3f(x)g(x) = (3) (lim_x→3 f(x)) (lim_x→3 g(x)) = (3) (4) (-2)) = -24

(e) lim_x→3 g(x)h(x) = (lim_x→3 g(x)) (lim_x→3 h(x)) = (−2)(0) = 0

(f) lim_x→3 g(x)h(x)f(x) = (lim_x→3 g(x)) (lim_x→3 h(x)) (lim_x→3 f(x)) = (−2) (0)(4) = 0