Let f(x)=x2+x−6x−2. Note that f(x) is undefined at x=2. However, in this case, we can find a continuous function g(x) defined at x=2 that agrees with f(x) near x=2. Find the function g(x) that is equal to f(x) when x≠2 such that
g(2)=limx→2f(x).
You must have a typo.
I read your function as
f(x) = x^2 - 5x - 2, which is defined for all values of x
to be undefined there must have been a denominator of (x-2)
it's f(x)=x^2+x−6 / x−2
and the answer is: x + 3
To find a continuous function g(x) that agrees with f(x) near x=2, we can start by evaluating the limit of f(x) as x approaches 2.
First, let's find the limit:
limx→2 f(x)
To evaluate this limit, we can simplify the function f(x) by factoring the numerator:
f(x) = x^2 + x - 6x - 2
= x^2 - 5x - 2
Next, we can substitute x=2 into the simplified function to find the value of the limit:
limx→2 (x^2 - 5x - 2) = 2^2 - 5(2) - 2 = 4 - 10 - 2 = -8
Therefore, the limit as x approaches 2 is equal to -8. Now, to find the function g(x) that agrees with f(x) near x=2, we can define it as:
g(x) = {
f(x) if x ≠ 2,
limx→2 f(x) if x = 2
}
Since we found that limx→2 f(x) = -8, we can substitute this value into the definition of g(x):
g(x) = {
x^2 - 5x - 2 if x ≠ 2,
-8 if x = 2
}
Therefore, the function g(x) that is equal to f(x) when x ≠ 2 and g(2) = limx→2 f(x) is:
g(x) = {
x^2 - 5x - 2 if x ≠ 2,
-8 if x = 2
}