Question

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 true :

lim (x,y)→(0,0) (3x − y) ^4 + 2x^2 + 2y^2 / x ^2 + y^2 = 2.

could you please use the formal limit definition above ?

Thank

Answer #1

**For
any doubt please let me know in comment box. Thank you**

Assumptions:
The formal definition of the limit of a function is as follows:
Let ƒ : D →R with x0 being an
accumulation point of D. Then ƒ has a limit L at
x0 if for each ∈ > 0 there is a δ > 0
that if 0 < |x – x0| < δ and
x ∈ D, then |ƒ(x) – L| <
∈.
Let L = 4P + Q. when P = 6 and Q = 24
Define...

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.

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lim(x,y)→(0,0)5x^2/2x^2+y^2
1) Along the xx-axis:
2) Along the yy-axis:
3) Along the line y=mxy=mx :
4) The limit is:

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Use Deﬁnition 4.2.1 to supply a proper proof for the following
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(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|...

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f(x, y, z) = sin z, x ≥ 0, y ≥ 0, and below the plane 2x + 2y +
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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
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(c) Show that f(x; y) has no limits as (x; y) approaches (0;
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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...

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• 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,...

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