A function f(x) is said to have a removable discontinuity at x=a if:

1. f is either not defined or not continuous at x=a.
2. f(a) could either be defined or redefined so that the new function IS continuous at x=a.

Let f(x)=2x^2+3x–14/x–2
Show that f(x) has a removable discontinuity at x=2 and determine what value for f(2) would make f(x) continuous at x=2.

f(2) = ?

To determine if the function f(x) has a removable discontinuity at x = 2, we need to analyze its behavior around this point.

Let's start by finding the value of f(2):

f(x) = (2x^2 + 3x - 14) / (x - 2)

To find f(2), we substitute x = 2 into the function:

f(2) = (2(2)^2 + 3(2) - 14) / (2 - 2)
= (8 + 6 - 14) / 0
= 0 / 0

As we can see, f(2) is undefined since we get a division by zero. This indicates that there may be a point of discontinuity at x = 2.

To make f(x) continuous at x = 2, we need to redefine f(2) in a way that eliminates the division by zero issue. This can be done by finding the limit of f(x) as x approaches 2 and assigning that value to f(2).

Now, let's find the limit of f(x) as x approaches 2:

lim(x->2) (2x^2 + 3x - 14) / (x - 2)

We can simplify the expression by factoring the numerator:

lim(x->2) [(x - 2)(2x + 7)] / (x - 2)

Notice that (x - 2) cancels out:

lim(x->2) (2x + 7)

Evaluating the limit as x approaches 2:

lim(x->2) (2x + 7) = 2(2) + 7 = 4 + 7 = 11

Therefore, we have found that the limit of f(x) as x approaches 2 is 11.

To make f(x) continuous at x = 2, we can assign f(2) a value of 11:

f(2) = 11

By redefining f(2) as 11, the new function becomes:

f(x) = (2x^2 + 3x - 14) / (x - 2), x ≠ 2
= 11, x = 2

This new function is now continuous at x = 2, and the removable discontinuity has been addressed.

To show that the function f(x) has a removable discontinuity at x=2, we need to check if the function is undefined or discontinuous at x=2.

To do this, let's evaluate the function f(x) at x=2.

f(2) = (2(2)^2 + 3(2) - 14)/(2 - 2)
= (2(4) + 6 - 14)/(0)
= (8 + 6 - 14)/(0)
= 0/0

Notice that the denominator becomes zero, which means f(x) is undefined at x=2. Therefore, f(x) is not defined at x=2.

Now, to determine the value for f(2) that would make f(x) continuous at x=2, we can redefine the function at x=2.

In order to remove the discontinuity and make the function continuous at x=2, we can take the limit of f(x) as x approaches 2.

lim(x->2) [(2x^2 + 3x – 14)/(x – 2)]

By using algebraic manipulation, we can simplify the expression:

lim(x->2) [(2x^2 + 3x – 14)/(x – 2)]
= lim(x->2) [(2x + 7)(x – 2)/(x – 2)] (factoring numerator)
= lim(x->2) [(2x + 7)]
= 2(2) + 7 (substituting x=2)
= 4 + 7
= 11

Therefore, in order to make f(x) continuous at x=2, we need to redefine f(2) as 11.

So, f(2) = 11 would make f(x) continuous at x=2.