Consider the following.
lim x→−2
2x^2 − 2x − 12/x + 2
Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.)
Write a simpler function that agrees with the given function at all but one point.
g(x) = ??? thnks
To find the limit of the given function as x approaches -2, we can directly substitute -2 into the function:
lim x→-2 (2x^2 - 2x - 12) / (x + 2)
By substituting -2 into the function, we get:
(2*(-2)^2 - 2*(-2) - 12) / (-2 + 2)
= (2*4 + 4 - 12) / 0
Since the denominator is 0, we cannot divide by 0. Thus, the limit does not exist (DNE).
To write a simpler function that agrees with the given function at all but one point, we can remove the singularity at x = -2. We can accomplish this by canceling out the common factor of (x + 2) in the numerator and denominator:
g(x) = (2x^2 - 2x - 12) / (x + 2) * (x + 2) / (x + 2)
Simplifying this expression, we get:
g(x) = (2x^2 - 2x - 12) / (x + 2)
Now, g(x) is a simpler function that agrees with the given function at all points except x = -2.