Question

Suppose that f : R → R such that, the lim h→0 [f(x) − f(x − h)] = 0 for all x ∈ R, then is f continuous in this case?

Answer #1

The derivative of a function, f(x) is equal to lim h->0
f(x+h)-f(x) / h. True or false. please explain.

f(x) =x2 -x
use f'(x)=lim h->0 f(x+h) - f(x)/h
find:
1. f '(x)
2. f '(2)
3. Find the equation of a tangent line to the given function at
x=2
4. f ' (-3)
5. Find the equation of a tangent line to the given function at
x=-3

consider the function
f(x)= -1/x, 3, √x+2
if x<0
if 0≤x<1
if x≥1
a)Evaluate lim,→./ f(x) and lim,→.2 f(x)
b. Does lim,→. f(x) exist? Explain.
c. Is f(x) continuous at x = 1? Explain.

Let f(x)=7x^2+7. Evaluate
lim h→0 f(−1+h)−f(−1)/h
(If the limit does not exist, enter "DNE".)
Limit =

A function f”R n × R m → R is bilinear if for all x, y ∈ R n and
all w, z ∈ R m, and all a ∈ R: • f(x + ay, z) = f(x, z) + af(y, z)
• f(x, w + az) = f(x, w) + af(x, z) (a) Prove that if f is
bilinear, then (0.1) lim (h,k)→(0,0) |f(h, k)| |(h, k)| = 0. (b)
Prove that Df(a, b) · (h, k) = f(a,...

Prove or give a counter example: If f is continuous on R and
differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f
is differentiable on R .

Suppose that r is a double zero of the C2 function f, i.e., f(r)
= f′(r) = 0 but f′′(r) is not 0. Show that Newton’s method applied
to f converges linearly with the asymptotic constant 1/2, i.e.,
show that
lim n->infinity | x(n+1)−r | / | x(n)−r | = 1/2.

1. Let f : R2 → R2, f(x,y) = ?g(x,y),h(x,y)? where g,h : R2 → R.
Show that
f is continuous at p0 ⇐⇒ both g,h are continuous at p0

If f:R→R satisfies f(x+y)=f(x) + f(y) for all x and y and
0∈C(f), then f is continuous everywhere.

Consider the expression lim as x→∞ of g(x)^f(x). Suppose we
know
lim as x→∞ of g(x) = 1
lim as x→∞ of f(x) = ∞
Explain using sentences (that can include mathematical symbols
and expressions) how you would approach evaluating this limit.

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