Find the value of a that will make h(x) continuous

Question

2x²+ax+3 ; x < -4

3x+a ; x >= -4

Answer 1

First find the appropriate limits:

lim_{x→-4 -} (2x² + ax + 3) = 2(-4)² - 4a + 3 = 35 - 4a

lim_{x→-4 +} (3x + a) = 3(-4) + a= a - 12

The two limits must match in order to make the function continuous.

a - 12 = 35 - 4a

5a = 47

a = 47/5

Answer 2

For the function to be continuous, we would need 1) the limit of the function to exist at the boundary of the two intervals (at x = -4) AND 2) we would need that limit to equal the value of the function at that point. That is, the limit would need to equal f(-4) = 3(-4) +a = -12-a.

For the limit to exist, we need the limits as x approaches -4 from the left and right sides to match:

lim_x→-4+= 3(-4)+a

lim_{x→-4- =} 2(-4)²+a(-4)+3

Setting equal gives

2(-4)²+a(-4)+3 = 3(-4)+a

2(16)-4a+3 = -12+a

35-4a = -12+a

-5a = -47

a = 47/5

Note that this value also ensures that lim_x→-4f(x) = f(-4) as desired. So, when a = 47/5, the function would be continuous.