Question

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.

Answer #1

The following statement is FALSE: If lim x!6 [f(x)g(x)] exists,
then the limit must be f(6)g(6).
Give an example of two functions f(x) and g(x) that demonstrate
the falsity of this state- ment - that is, two functions f(x) and
g(x) such that lim x!6 [f(x)g(x)] exists, but is not equal to
f(6)g(6). Explain your answer.

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.

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.)
=

hi guys , using this definition for limits in higher dimensions
:
lim (x,y)→(a,b) f(x, y) = L
if 1. ∃r > 0 s.th. f(x, y) is defined when 0 < || (x, y) −
(a, b) || < r
and 2. given ε > 0 we can find δ > 0 s.th. 0 < || (x,
y) − (a, b) || < δ =⇒ | f(x, y) − L | < ε
how do i show that this is...

Let f(x) and g(x) be polynomials and suppose that we have f(a) =
g(a) for all real numbers a. In this case prove that f(x) and g(x)
have exactly the same coefficients. [Hint: Consider the polynomial
h(x) = f(x) − g(x). If h(x) has at least one nonzero coefficient
then the equation h(x) = 0 has finitely many solutions.]

1A. Complete the table. (Round your answers to five decimal
places.)
lim x→0
x + 16
− 4
x
x
−0.1
−0.01
−0.001
0
0.001
0.01
0.1
f(x)
?
Use the result to estimate the limit. Use a graphing utility to
graph the function to confirm your result. (Round your answer to
five decimal places.)
lim x→0
x + 16
− 4
x
≈
1B.
Find the limit L.
lim x→−6
2x2 + 16x + 24
x + 6
L...

1. Evaluate the limit using L'Hospital's rule if necessary
lim x→∞ (1+ 12 / x)^x/1
2. In which limits below can we use L'Hospital's Rule?
lim x→π/5 sin(5x) /5x−π
lim x→−∞ e^−x / x
lim x→0 2x/ cotx
lim x→0 sin(3x) / 3x
I Need help with both questions please! thank you so much.

Consider the following expression: 7^n-6*n-1
Using induction, prove the expression is divisible by 36.
I understand the process of mathematical induction, however I do
not understand how the solution showed the result for P_n+1 is
divisible by 36? How can we be sure something is divisible by 36?
Please explain in great detail.

Consider the following limit.
lim (x^2 + 4)
(x--> 5)
1. Find the limit L.
2. Find the largest δ such that |f(x) −
L| < 0.01 whenever 0 < |x − 5| < δ.
(Assume 4 < x < 6 and δ > 0. Round your answer to
four decimal places.)
I am honestly so lost... if you could please show work I would
greatly appreciate it!!

1. Evaluate the limit
lim x→0 sin4x /7x
2. In which limits below can we use L'Hospital's Rule? select
all that apply.
lim x→∞ e^−x /x
lim x→0 1−e^x /sin(2x)
lim x→0 x+tanx /sinx
lim x→−∞ e^−x /x

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