Question

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.

Answer #1

2.
Consider the following function.
f(x) =
x2 + 9
if x ≤ 1
5x2 − 1
if x > 1
Find each value. (If an answer does not exist, enter DNE.)
f(1)=
lim x→1− f(x)
=
lim x→1+ f(x)
=
Determine whether the function is continuous or discontinuous at
x = 1.
Examine the three conditions in the definition of
continuity.
The function is continuous at x = 1.?
or
The function is discontinuous at x = 1. ?

Is the function f(x,y)=x−yx+y continuous at the point (−1,−1)?
If not, why is the function not continuous?
Select the correct answer below:
A. Yes
B. No, because lim(x,y)→(−1,1)x−yx+y=−1 and f(0,0)=0.
C. No, because lim(x,y)→(−1,1)x−yx+y does not exist and f(0,0)
does not exist.
D. No, because lim(x,y)→(0,0)x2−y2x2+y2=1 and f(0,0)=0.

a.)Consider the function f (x) = 3x/ x^2 +1
i) Evaluate f (x+1), and f (x)+1. Explain the difference. Do the
same for f (2x) and 2f (x).
ii) Sketch y = f (x) on the interval [−2, 2].
iii) Solve the equations f (x) = 1.2 and f (x) = 2. In each
case, if a solution does not exist, explain.
iv) What is the domain of f (x)?
b.)Let f (x) = √x −1 and g (x) =...

Consider the function given byf(x, y)
=((y^2−2^x)/(2x^2+y^2))^3.Show that lim(x,y)→(0,0)f(x, y)does not
exist by computing the limit along the positivex-axis and the
positivey-axis.

1) Consider the function.
f(x) = x5 − 5
(a) Find the inverse function of f.
f −1(x) =
2)
Consider the function
f(x) = (1 + x)3/x.
Estimate the limit
lim x → 0 (1 + x)3/x
by evaluating f at x-values near 0. (Round
your answer to five significant figures.)
=

Consider a function f(x; y) =
2x2y
x4 + y2 .
(a) Find lim
(x;y)!(1;1)
f(x; y).
(b) Find an equation of the level curve to f(x; y) that passes
through the point (1; 1).
(c) Show that f(x; y) has no limits as (x; y) approaches (0;
0).

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

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

consider the function f(x)=-x2+90x-30
1)determine f'(x) (use lim method)
2)determine the equation of the line tangent to f(x) at
x=1

Consider the function f(x)=xarctan(1/x) if x≠ 0 ,0 if x = 0. (1)
Using the defnition of continuity at a point you studied in
Calculus 1 check whether f(x) is continuous at x = 0. (2) Using the
limit defnition of differentiability at a point check whether f'(0)
exists.

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