In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
The limit inferior of a sequence
x
n
{\displaystyle x_{n}}
is denoted by
lim inf
n
→
∞
x
n
or
lim
_
n
→
∞
x
n
.
{\displaystyle \liminf _{n\to \infty }x_{n}\quad {\text{or}}\quad \varliminf _{n\to \infty }x_{n}.}
The limit superior of a sequence
Apostol defines limit for vector fields as
> ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)##
means that :
##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0##
Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##
I do not understand how they got the -x in the numerator to turn into a sqrt(x) when factoring to solve this multivariable function. Could some help me understand?
To an average person with high school math knowledge how would you explain in a few words why no object could travel faster than the speed of light ?
Well it's because...
Use a graph to investigate limit of f(x) as
x→c at the number c.
Note: c is given to be 2. This number comes from the side conditions of the piecewise function.
See attachments.
lim (x + 2) as x tends to c from the left is 2.
lim x^2 as x tends to c from the right is 4.
LHL does not...
Use the graph to investigate the limit of f(x) as x tends to c at the number c.
See attachments.
Based on the graph of f(x), here is what I did:
lim (2x + 1) as x tends to 0 from the left is 1.
lim (2x) as x tends to 0 from the right is 0.
LHL does not equal RHL.
I conclude the limit of...
Use the graph to investigate
(a) lim of f(x) as x→2 from the left side.
(b) lim of f(x) as x→2 from the right side.
(c) lim of f(x) as x→2.
Question 20
For part (a), as I travel along on the x-axis coming from the left, the graph reaches a height of 4. The limit is 4. It does not matter...
Use the graph to investigate
(a) lim of f(x) as x→2 from the left side.
(b) lim of f(x) as x→2 from the right side.
(c) lim of f(x) as x→2.
Question 18
For part (a), as I travel along on the x-axis coming from the left, the graph reaches a height of 4. The limit is 4. It does not matter if...
My apologies. I posted the correct problem with the wrong set of instructions. It it a typo at my end. Here is the correct set of instructions for 28:
Use the graph to investigate limit of f(x) as x→c. If the limit does not exist, explain why.
For (a), the limit is 1.
For (b), the limit DOES...
For questions 24 and 26, Use the graph to investigate limit of f(x) as x→c. If the limit does not exist, explain why.
Question 24
For (a), the limit is 1.
For (b), the limit is cannot be determined due to the hole at (c, 2).
For (c), LHL does not = RHL.
I conclude the limit does not exist...
Investigate A Limit
Investigate the limit of f(x) as x tends to c at the given c number.
Attachment has been deleted.
Let me see.
Let c = 2
I think I got to take the limit of f(x) as x tends to 2 from the left and right. What about as x tends to 2 (from the left and right at the same...
Summary:: Graphs and Limits
Use the graph to determine the limit of the piecewise function as x tends to 1.
Let me see.
lim of (-x + 3) as x-->1 from the left is 2.
lim of (2x) as x-->1 from the right is 2.
I can safely say that the limit of f(x) as x tends to 1 from the left and right...
Summary:: Use Graph To Investigate Limit
Use the graph to investigate the limit of f(x)
as x tends to 0.
Let me see.
I got to use the graph to investigate the limit of f(x) as x tends to 0 from the left and right.
Let y = f(x).
The given function can also be expressed as f(x) = | x |.
The...
Hello everyone. How are you? I want to learn calculus so badly. I plan to do a self-study of calculus l, ll, and lll. Before thinking so far ahead, I need a clear, basic definition of the concept of a limit. Textbook language is never easy to grasp unless the student is gifted. I am not gifted...
If you are told something holds if the limit exists, and given a function f (specifically not piecewise defined), is it enough to show that the limit as x approaches c = the function evaluated at c?
With a piecewise defined function, it is easy to check both sides of a potential discontinuity...
I have the following definition:
$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$
From this, how can I get the definition of
$$\lim_{x\to p^-}=-\infty? $$
I have the following definition:
$$\lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$
From this, how can I get the definition of
$$\lim_{x\to p^-}=-\infty? $$
Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##.
Proof: First we'll regard the case ## l = \infty ## ( the case...
Hello!
I need to calculate the limit of this function ## f(x) = (x^2-9)*e^{-x}## for + and - ## \infty ## Now for + infinity I did this
$$ \frac{(x^2-9)}{e^x} $$ apply L'Hospital since we have infinity divided by infinity; $$\frac{2x}{e^x} $$ Apply L'Hospital again $$ \frac{2}{e^x} $$ the...
Hello, I know I posted this question recently but I wanted to update with my progress. I have figured out what the limit should be but I would really appreciate help with how to use the definition of the limit of a sequence to prove it! What I have is:Suppose n is extremely large, then both...
Hello! I have been trying to work through this but I have never really been able to use the definition correctly to find a limit sequence. Any help would be greatly appreciated!
Bernard Stiegler said that technology is an evolving organism that never ends as long as their are people; yet as an aspiring futurist; I feel that I’ve reached the limit to all the different concepts for technology. All the futurism stuff is repetitive because their are only so many things a...
Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##.
Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
height from ground
speed
100
40
80
48
60
60
40
72
20
80
I tried to plot the points (speed on x-axis and height on y axis) and I got more or less like a straight line but I am not sure whether the graph would help to calculate the upper and lower limit of the time.
I also tried to...
My attempt:
\begin{align}
\lim\limits_{n \to \infty} \sqrt{n^2 + n} - n &= n\sqrt{1+\frac{1}{n}} -n\\
&=n - n\\
&= 0\\
\end{align}
I think the issue is at (1)-(2)
For comparison, here is Rudin's solution
How would I determine the following limit without substitution of large values of x to see what value is approached by the complex function?
## \lim_{x \rightarrow +\infty} {\dfrac {2^{x}} {x^{2} } } ## where ## x\in \mathbb{R}##
Use the epsilon-delta method to show that the limit is 3/2 for the given function.
lim (1 + 2x)/(3 - x) = 3/2
x-->1
I want to find a delta so that | x - 1| < delta implies |f(x) - L| < epsilon.
| (1 + 2x)/(3 - x) - (3/2) | < epsilon
-epsilon < (1 + 2x)/(3 - x) - 3/2 < epsilon
I now add...
The model below is given to find the growth of a population of an endangered species.
P(t) = (500)/[1 + 82.3e^(-0.162t)]
Find the limit of P(t) as t tends to positive infinity.
The answer in the textbook is 500.
Can a model like this be graphed? If so, is the graph of P(t) the best approach...
Find the limit of (5x)/(100 - x) as x tends to 100 from the left side.
The side condition given: 0 <= x < 100
To create a table, I must select values of x slightly less than 100.
I did that and ended up with negative infinity as the answer. The textbook answer is positive infinity.
Can you...
Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to 0 from the right side.
The answer is u_0. How is the answer found? Seeking a hint or two.
Can this Law of Cooling be graphed? If so, what does the graph look like?
Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to positive infinity.
The answer is T. How is the answer found? Seeking a hint or two.
Find the limit of csc(2x) as x tends to pi/2 from the right side.
I decided to graph the function. Based on the graph, I stated the answer to be positive infinity.
According to the textbook, the answer is negative infinity. Why is negative infinity the right answer?
Thanks
Find the limit of cot (x) as x tends to pi from the left side.
Seeking a hint or two. Does the graph of the given function help in terms of finding the limit?
Find the limit of (1 - x)/[(3 - x)^2] as x---> 3.
I could not find the limit using algebra. So, I decided to graph the given function.
I can see from the graph on paper that the limit is negative infinity.
How is this done without graphing?
Find the limit of 1/(x^2 - 9) as x tends to -3 from the left side.
Approaching -3 from the left means that the values of x must be slightly less than -3.
I created a table for x and f(x).
x...(-4.5)...(-4)...(-3.5)
f(x)... 0.088...0.142...…...0.3076
I can see that f(x) is getting larger and...
Find the limit of 5/(x^2 - 4) as x tends to 2 from the right side.
Approaching 2 from the right means that the values of x must be slightly larger than 2.
I created a table for x and f(x).
x...2.1...2.01...2.001
f(x)...12...124.68...1249.68
I can see that f(x) is getting larger and larger...
Find the limit of (3x)/(x - 2) as x tends to 2 from the left side.
Approaching 2 from the left means that the values of x must be slightly less than 2.
I created a table for x and f(x).
x...0...0.5...1...1.5
f(x)...0...-1...-3...-9
I can see that f(x) is getting smaller and smaller and...
Find the limit of x/(x^2 - 4) as x tends to 2 from the right.
If I plug x = 0, I will get 0/-4 = asymptote. Again, is graphing the best to do this one?
I can also create a number line.
<----------(-2)----------(0)---------------(2)-------->
I can then select values for x from each interval...
Find the limit of (2x + 1)/(x + 4) as x tends to - 4 from the right side.
I know there's a vertical asymptote at x = -4. I think the best way to solve this problem is by graphing the function. I am not too sure about how to solve algebraically.
I am thinking about the number line...
Greetings!
In statistical mechanics, when studying diffusion processes, one often finds the following reasoning:
Suppose there is a strictly positive differentiable function ##f: \mathbb{R} \rightarrow \mathbb{R}## with ## \lim_{x \rightarrow +\infty} {f'(x)} = a > 0##.
Then for sufficiently...
Hello.
Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist.
But can a quotient of the two acutally approach a certain value?
lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to...
I tried taking e^ln but to no avail. Please help! Thanks.
My attempt:
$$\lim_{x\to 0^+}(1-\cos (\sqrt x))^{\sin(x)}$$
$$\lim_{x\to 0^+}e^{\ln (1-\cos\sqrt x)^{\sin x}}$$
$$\lim_{x\to 0^+}e^{{\sin x}\ln (1-\cos\sqrt x)}$$
$$\lim_{x\to 0^+}\exp(\frac{\ln (1-\cos\sqrt x)}{1/\sin x})$$
If I apply...
The speed limits on a straight road are given by a known function g(x,t) where x is the location on the road and t is time. A car starts at x = 0 at time t= 0 and always drives at the speed limit. The location of the car is given by the (unknown) function s(t). Is there a differential...
I wonder if the limit of the following can be converted into integral or some elegant form as N tends to infinity:
\[ \sum_{n=0}^{N}\frac{a}{2^{n}}\sin^{2}\left(\frac{a}{2^{n}}\right) \]
If we plot or evaluate the value then it does appear that the series converges very fast...
Hi all,
I'm a little confused about something.
Force-extension graphs and stress-strain graphs are always both straight lines up until the limit of proportionality, implying both the spring constant and the Young modulus are constant up until then.
For a force-extension graph, Hooke's Law...
Can someone please tell me how to solve a limit problem like this?
$$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$
This is my attempt to solve the problem:
$$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$
$$= \lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} -...
Hi guys, I am having difficulties in solving this limit.
Below, I'll attach my procedure which ends up in the indeterminate form ##0\cdot \infty##...
How could I solve it?
$$\lim_{x \to +\infty}(\sqrt[3]{x^3-4x^2}-x) \rightarrow
\lim_{x \to +\infty}(x\sqrt[3]{1-\frac{4}{x}}-x) \rightarrow...
Large wind turbines have become very efficient and have a power coefficient close to that defined by Betz.
However, large wind turbines are stopped when the wind is too strong, not because they produce too much, but because their blades are subject to bending stresses which may break their...
I have to prove that \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]= a (in order to show that the indeterminate form of the type 0^0 can be any positive real number).
This is what I did:
Let y = \lim_{x \rightarrow 0^+} \left[x^\left[(\ln a)/(1+ \ln x)\right] \right]...
Hi, PF
This is the quote:
"If ##m## is an integer and ##n## is a positive integer, then
6. Limit of a power:
## \displaystyle\lim_{x \to{a}}{\left[f(x)\right]^{m/n}} ## whenever ##L>0## if ##n## is even, and ##L\neq{0}## if ##m<0##"
What do I understand?
-whenever ##L>0## if ##n## is even: ##m##...
I'll write my considerations which lead me to get stuck on the ##\infty-\infty## form.
$$\lim_{x \to +\infty }\sqrt{x^{2}-2x}-x+1 \rightarrow |x|\sqrt{1-0}-x+1$$
And I have no idea on how to go on...