For what value of c is the function f(x) = when?

Question

c x = -2

3 x = 1

x^2-4/(x+2)(x-3) Otherwise

continuous at x = -2?

Answer 1

Continuity requires that as x get close to -2, then f(x) gets close to c. In the piecewise function, there are two pieces that define the value of the function "near" -2. The first piece defines it exactly at -2 to be c. The second piece, the last expression, defines it near -2 but not at -2 because that would make the denominator zero. Given that equality is as close as two numbers can be, you set the two expressions equal to each other, reduce and solve for x = -2.

Answer 2

What must be the value of constant C so that the function is continuous at x=-2

The numerator factor (x+2)(x-2)/[(x+2)(x-3)] =

x-2 / (x-3)

So (-2 -2)/(-2 -3) = -4/-5 = 4/5