Sage-Tips

Is the function f continuous at 0 0 )? Justify your answer?

Limit and Continuity : (i) We say that L is the limit of a function f : R3 → R at X0 ∈ R3 (and we write limX→X0 f(X) = L) if f(Xn) → L whenever a sequence (Xn) in R3, Xn = X0, converges to X0. Hence f is continuous at (0,0). In fact, this function is continuous on the entire R2.

Is the function f x/y differentiable at 0 0 )? Explain?

In particular, this value is not 0, so the original limit could not be equal to zero. (In fact, that limit does not exist though you don’t need to show that, just that it does not equal zero.) Therefore, f is not differentiable at (0,0).

How do you know if F is continuous at 0?

To prove that f is continuous at 0, we note that if 0 ≤ x<δ where δ = ϵ2 > 0, then |f(x) − f(0)| = √ x < ϵ. f(x) = ( 1/x if x ̸= 0, 0 if x = 0, is not continuous at 0 since limx→0 f(x) does not exist (see Example 2.7).

Does there exist a continuous function f 0 1 → 0 ∞ which is onto?

Yes. Observe that there is an injective function from [0,1] to [0.1). (For example, f(x)=x/2.) There is also an injective function from [0,1) to [0,1].

Does lim x -> 1 f/x exist?

The limit of f as x approaches 1 exists and is 1, as f approaches 1 from both the right and left. Therefore limx→1f(x)=1.

For what value of A does lim x → ax exist?

lim ⁡ x → a [ x ] exists when a is not an integer.

Is the function f x/y xy differentiable at 0 0 justify your answer ))?

(c) f is not differentiable at (0,0). Solution. As (x, y) → (0,0), the first term in this product goes to 0, while the second is bounded. Thus, lim(x,y)→(0,0) f(x, y) = 0, and the function is continuous.

Is the function f x/y differentiable at every point?

Not every function is differentiable at every number in its domain even if that function is continuous. For example f(x) = |x| is not differentiable at 0 but f is continuous at 0.

What is the epsilon delta definition of a limit?

Epsilon-Delta Definition of a Limit. \\delta δ definition of a limit is an algebraically precise formulation of evaluating the limit of a function. Informally, the definition states that a limit. L L.

What does εvarepsilonε-δdeltaδ mean?

In calculus, the εvarepsilonε-δdeltaδ definition of a limit is an algebraically precise formulation of evaluating the limit of a function. Informally, the definition states that a limit LLL of a function at a point x0x_0x0​ exists if no matter how x0x_0 x0​ is approached, the values returned by the function will always approach LLL.

How do you prove a limit using the ε\\varepsilonε-δ\\deltaδ technique?

In general, to prove a limit using the ε\\varepsilonε-δ\\deltaδ technique, we must find an expression for δ\\deltaδ and then show that the desired inequalities hold. The expression for δ\\deltaδ is most often in terms of ε,\\varepsilon,ε, though sometimes it is also a constant or a more complicated expression.