f(x)=x/e^x
Show that f is continuous and differentiable for all real numbers.
since e^x is never zero, it is clear that f(x) is continuous everywhere
so, what does it mean to be differentiable?
I believe it implies that the function can be differentiated
yes. That involves evaluating a limit. If that limit exists, then the function is differentiable.
To show that the function f(x) = x/e^x is continuous and differentiable for all real numbers, we need to demonstrate two things:
1. Continuity: We need to show that the function is continuous at every point in its domain, which is all real numbers.
2. Differentiability: We need to show that the function is differentiable at every point in its domain.
Let's start by proving the continuity of f(x) using the definition of continuity.
Definition of Continuity:
A function f(x) is continuous at a point a if the following three conditions are met:
1. f(a) is defined.
2. The limit of f(x) as x approaches a exists.
3. The limit of f(x) as x approaches a is equal to f(a).
Now, let's verify these conditions for the function f(x) = x/e^x.
1. f(a) is defined:
The function f(x) = x/e^x is defined for all real numbers. So, f(a) is defined for every real number a.
2. The limit of f(x) as x approaches a exists:
To determine the limit as x approaches a, we'll have to evaluate the limit of x/e^x as x approaches a.
Using L'Hôpital's rule:
Apply L'Hôpital's rule by taking the derivative of the numerator and denominator successively until an indeterminate form (0/0 or ∞/∞) is reached.
Let's differentiate the numerator and denominator:
f(x) = x/e^x
Differentiating the numerator:
f'(x) = 1
Differentiating the denominator:
f''(x) = e^x
As x approaches a, the limit of f'(x) is 1, and the limit of f''(x) is e^a. Since both limits exist, the limit of f(x) as x approaches a exists.
3. The limit of f(x) as x approaches a is equal to f(a):
Now, we need to check if the limit of f(x) as x approaches a is equal to f(a) for every a.
Taking the limit as x approaches a:
lim(x -> a) (x/e^x) = a/e^a
Therefore, the limit of f(x) as x approaches a is equal to a/e^a.
Since all three conditions for continuity are satisfied, we can conclude that f(x) = x/e^x is continuous for all real numbers.
To prove that f(x) = x/e^x is differentiable for all real numbers, we just need to show that the derivative of f(x) exists at every point in its domain.
We have already differentiated f(x) previously, and the derivative of f(x) is 1 - (x/e^x) = (e^x - x)/(e^x).
Since the derivative is defined for all real numbers, we can conclude that f(x) = x/e^x is differentiable for all real numbers.
In summary, we have shown that the function f(x) = x/e^x is continuous and differentiable for all real numbers by verifying the conditions of continuity and differentiability.