Question

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!!

Answer #1

we are given

**(1)**

we can plug x=5

**(2)**

we can plug x=5 on right side

now, we can compare with

0 < |*x* − 5| < δ

so, we get

**............Answer**

Consider the following limit.
lim x→4 (x2 + 8)
Find the limit L.
L = 24
(a) Find δ > 0 such that |f(x) − L| < 0.01 whenever 0 <
|x − c| < δ.
(Round your answer to five decimal places.)
δ =
(b) Find δ > 0 such that |f(x) − L| < 0.005 whenever 0 <
|x − c| < δ.
(Round your answer to five decimal places.)
δ =

Use Deﬁnition 4.2.1 to supply a proper proof for the following
limit statements.
(a) lim as x→2 of (3x + 4) = 10.
(d) lim as x→3 of 1/x =1 /3.
Deﬁnition 4.2.1 (Functional Limit). Let f : A → R, and let c be
a limit point of the domain A. We say that limx→c f(x)=L provided
that, for all ϵ>0, there exists a δ>0 such that whenever 0
< |x−c| <δ(and x ∈ A) it follows that |f(x)−L|...

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...

A graphing calculator is recommended.
For the limit
lim x → 2 (x3 −
2x + 5) = 9
illustrate the definition by finding the largest possible values
of δ that correspond to ε = 0.2 and ε =
0.1. (Round your answers to four decimal places.)

By using delta- epsilon show that the two definitions of the
limit are equivalent
Def1: lim┬(x→x_0 )〖f(x)=f(x_0)〗. If for any ϵ>0,there exist
a δ>0 such that 0<|x-x_0 |<δ implies |f(x)-L|<ϵ
Def2: If for any sequence {x_n }→x_0 we have f(x_n)→L

a.) Find the following limit.
lim x→−∞ x^3 - sqrt(4x^6-3x)/7x^3+x
b.) Sketch the graph of a function f(x) that has all of the
following features:
f ' (x) > 0 on the intervals (−∞, −4) ∪ (3,∞).
f ' (x) < 0 on the intervals (−4, −1) ∪ (−1, 3).
f '' (x) > 0 on the intervals (−∞, −4) ∪ (−4, −2) ∪ (−1,
4).
f '' (x) < 0 on the intervals (−2, −1) ∪ (4,∞).
x-intercepts when...

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...

Please find a formula that meet those 5 conditions:
1. lim x→2+ f(x) = ∞
2. lim x→2− f(x) = −∞
3. f(1) = 0
4. f(0) = 3
5. limx→∞ f(x) = 0

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.

Limits Analytically
3) Calculate the following limit . showing all work for full
credit!
lim √x+h+4 - √x+4 / h
h--> 0
4) Use algebra and the fact learned about the limits of sin x /
x to calculate the following limit analytically, showing all
work!
*note* sin x / x = 1 as x-->0
solve:
lim sin(2L) / sin (5L)
L--> 0

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