Consider the following fucntion

Question

g(x) { x^2 + 5 if x < 1

-x + 7 if 1 ≤ x ≤ 4

srt(x) - 2 if x ≥ 4

find lim x -> 1+ g(x)

find lim x -> 1- g(x)

find lim x -> 1 g(x)

find lim x -> 4+ g(x)

find lim x -> 4- g(x)

find lim x -> 4g(x)

Answer 1

It helps to see the graph (my guess is that one of the defined intervals does not have an or equal to symbol on the 4):

desmos.com/calculator/6romricohl

lim_x->1+ g(x) = lim_x->1+ (-x+7) = -1 + 7 = 6 (Use the 2nd rule because numbers like 1.001 are in the interval from 1 to 4 inclusive.

lim_x->1- (x²+5) = 6

Since limit from the left equals limit from the right the lim_x->1g(x) = 6.

But a different story for 4:

lim_x->4+(√4 - 2) = 0

lim_x->4-(-x+7) = 3

Since limit from the left does not equal limit from the right the limit as x ->4 does not exist.