Let f(x)=(2+x)2−4x. Note that f(x) is undefined at x=0. However, in this case, we can find a continuous function g(x) defined at x=0 that agrees with f(x) near x=0. Find the function g(x) that is equal to f(x) when x≠0 such that
g(0)=limx→0f(x).
To find the function g(x) that agrees with f(x) near x=0, we need to find the limit of f(x) as x approaches 0.
First, let's calculate the limit of f(x) as x approaches 0:
lim(x→0) f(x) = lim(x→0) [(2+x)^2 - 4x]
To evaluate this limit, we can simplify the expression by expanding the square term:
lim(x→0) [(4 + 4x + x^2) - 4x]
= lim(x→0) (4 + x^2)
Now, in order to make this limit well-defined at x=0, we can replace x^2 with a function that is equal to x^2 for all x ≠ 0 but is equal to 0 at x=0. This will ensure continuity at x=0.
So, we can define g(x) as:
g(x) = 4 + x^2, for x ≠ 0
g(0) = lim(x→0) f(x) = 4
Therefore, the function g(x) that agrees with f(x) near x=0 and satisfies g(0) = lim(x→0) f(x) is:
g(x) = 4 + x^2, for all x.