Question

Choose the value of k that makes the following function continuous at x=1: Help!?

f(x) =

1. -8x^2+48x-40/(x-1) x is less than 1

2. -2x+k x is greater than or equal to 1

Answer 1

`k = 34`

Explanation:

In order for `f(x)` to be continuous we must have:

`lim_(x->1) f(x) = f(1)`

Now for the limit to exist in the first place, we must have:

`lim_(x->1^+) f(x) = lim_(x->1^-) f(x)`

When `x->1^+` from the right, `x > 1` then:

`lim_(x->1^+) f(x) = lim_(x->1^+) -2x+k = k-2`

while:

`lim_(x->1^-) f(x) = lim_(x->1^-) (-8x^2+48x-40)/(x-1)`

`lim_(x->1^-) f(x) = -8 lim_(x->1^-) (x^2-6x+5)/(x-1)`

`lim_(x->1^-) f(x) = -8 lim_(x->1^-) ((x-1)(x-5))/(x-1)`

`lim_(x->1^-) f(x) = -8 lim_(x->1^-) x-5`

`lim_(x->1^-) f(x) = 32`

Posing:

`k-2 = 32`

we get:

`k=34`