# How do you prove that the function f(x) = | x | is continuous at x=0, but not differentiable at x=0?

See the explanation, below.

Therefore,

Observe that

So the two sided limit does not exist.

By signing up, you agree to our Terms of Service and Privacy Policy

To prove that the function f(x) = |x| is continuous at x=0, we need to show that the limit of f(x) as x approaches 0 exists and is equal to f(0).

To do this, we consider the left-hand limit and the right-hand limit separately.

For the left-hand limit, as x approaches 0 from the left side (x < 0), f(x) = -x. Taking the limit as x approaches 0 from the left, we have lim(x→0-) -x = 0.

For the right-hand limit, as x approaches 0 from the right side (x > 0), f(x) = x. Taking the limit as x approaches 0 from the right, we have lim(x→0+) x = 0.

Since both the left-hand limit and the right-hand limit are equal to 0, we can conclude that the limit of f(x) as x approaches 0 exists and is equal to f(0), which is also 0. Therefore, f(x) = |x| is continuous at x=0.

To prove that f(x) = |x| is not differentiable at x=0, we need to show that the derivative of f(x) does not exist at x=0.

The derivative of f(x) can be found by considering the left-hand derivative and the right-hand derivative separately.

For the left-hand derivative, as x approaches 0 from the left side (x < 0), f'(x) = -1.

For the right-hand derivative, as x approaches 0 from the right side (x > 0), f'(x) = 1.

Since the left-hand derivative (-1) is not equal to the right-hand derivative (1), we can conclude that the derivative of f(x) does not exist at x=0. Therefore, f(x) = |x| is not differentiable at x=0.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you prove that the limit of #x^(-1/2) = 2# as x approaches 1/4 using the epsilon delta proof?
- For what values of x, if any, does #f(x) = 1/((x+8)(x-2)) # have vertical asymptotes?
- How do you prove that the limit of #((x/4)+3) = 9/2# as x approaches 6 using the epsilon delta proof?
- How do you find the limit of #1/(1+e^(1/x-2))# as x approaches #-2^+#?
- How do you evaluate the limit #(s-1)/(s^2-1)# as s approaches #1#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7