If a function f(x) is continuous, then what can be said about the value of the following limit
lim-->2 [f(x) +4x]
I figured this out this way but am not sure it's right.
f(2) + 4(2) = f(2) + 8. From this we can tell that it exists, but I'm not sure why.
If f(x) is continuous (on ℝ), then f(2) must exist. Also, it is known that polynomials are continuous on ℝ.
Within these constraints, limits are additive, so
Lim x->2 f(x)+4x
=Lim x->2 f(x) + Lim x->2 4x
=f(2) + 4(2)
=f(2) + 8,
as you have found.
To determine the value of the limit lim(x→2) [f(x) + 4x], we need to consider the continuity of the function f(x) at x = 2.
If a function f(x) is continuous at x = 2, it means that the value of f(x) remains close to the value at x = 2 as x approaches 2. In other words, for the limit as x approaches 2, we can substitute the value of f(x) as f(2) without affecting the result.
Therefore, we can rewrite the limit as:
lim(x→2) [f(x) + 4x] = lim(x→2) [f(2) + 4x]
We can now evaluate the new expression. The value of f(2) is known (assuming it is given or can be determined based on the properties of f(x)), and 4x is a linear term that approaches 8 as x approaches 2.
So, the value of the limit lim(x→2) [f(x) + 4x] is f(2) + 8.