Questions tagged [indeterminate-forms]
If the expression obtained after any substitution during limit analysis does not give enough information to determine the original limit, it is known as an indeterminate form.
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Why are we allowed to classify indeterminate limits with a specific form without proving it formally?
I have a few questions about infinite limits and their properties. Like I know that the arithmetic rules for the extended real number system are proven based on infinite limits, and those properties ...
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Edge case of L'Hopital's rule: is my reasoning correct?
In short: If we have a limit of the form $0/0$ and L'Hopital yields a limit that does not exist, due to a vertical asymptote with differing left and right limits. Does the same conclusion hold for the ...
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Evaluating the limit $\lim_{(x,y)\to(0,0)}\frac {x^3+y^3}{x^2+y^2}$
My task is to evaluate the following limit:
$$\lim_{(x,y)\to(0,0)}\frac {x^3+y^3}{x^2+y^2}$$
My attempts:
Let $x\neq 0$ and $y\neq 0$. Because in these paths the limit is equal to $0$. Suppose that $\...
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Since $y=x+2$ can be written $y=x^1 +2\cdot x^0$, do we have to accept that $0^0=1$ to plot the graph at $x=0$?
So when we graph, say, $y=x+2$, we could also write it as $y=x^1 +2\cdot x^0$. The point on the graph where $x=0$ has $y=2$, which mathematically speaking would be $y=0^2+ 2\cdot 0^0$. In order for ...
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Evaluating $\lim_{n \to \infty} \left( \frac{\left(1 + \frac 1n\right)^n}{\left(1 - \frac 1n \right)^n} - e^2\right)n^2$ [closed]
I am trying to find the limit
$$\lim_{n \to \infty} \left( \frac{\left(1 + \frac 1n\right)^n}{\left(1 - \frac 1n \right)^n} - e^2\right)n^2$$
I have been trying this but every time at the end I get ...
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Why cant we say that $\frac{\mathrm dy}{\mathrm dx}$ for every function is $\frac00$?
I’m a high school student trying to understand derivatives. According to me, $\frac{\mathrm dy}{\mathrm dx}$ represents what $\frac{\Delta y}{\Delta x}$ approaches as $\Delta x \to 0$. When $\Delta x =...
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Is $ \lim_{x \rightarrow \infty} \left[ \left( \frac{x+1}{4x-1} \right)^x \right] $ indeterminate form?
Given the limit
$$ \lim_{x \rightarrow \infty} \left[ \left( \frac{x+1}{4x-1} \right)^x \right] $$
Solve if possible, is there any indeterminate form?
This is a limits exercise on my calc I problem ...
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Evaluate $\lim_{t \to 0} \frac{3 \sin t- \sin 3t}{3 \tan t- \tan 3t}$.
The problem:
Evaluate $\displaystyle\lim_{t \to 0} \frac{3 \sin t- \sin 3t}{3 \tan t-\tan3t}$.
The solution:
$-\dfrac{1}{2}$
What I have tried:
$\displaystyle\lim_{t \to 0} \frac{3 \sin t- \sin 3t}{3 \...
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Problem with evaluating $\lim_{x\to-\infty}\sqrt{x^2-2x+3}+x$ by factoring $x^2$ from the radical [closed]
I have a question, excuse me if it is too simple for your understanding.
$$\begin{align*}
\lim_{x\to-\infty}\sqrt{x^2-2x+3}+x&=\infty-\infty\\
\lim_{x\to-\infty}\sqrt{x^2\left(1-\frac2x+\frac3{x^2}...
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Slopes of intersecting curves that are part of a relation
The relation $y^{(y^x)}-x=0$ consists of two curves that intersect at the point $P=(e^{-1},e^{-e})$. I'm interested in the slopes of these curves at this point. Differentiating the relation and ...
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Formula for limits of indeterminate forms of $1^\infty$
I'm currently learning limits now. I`ve reached the "$1^\infty$" case. During my search for online documents which in some way could help me tackle these limits I found two formulas:
$\lim_{...
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How to evaluate $\lim_{x \to 0} \,(x\ln^2x)$? [duplicate]
I was solving some Integral using contour technique where I needed to evaluate following limit:
$$\lim_{x \to 0} \,(x\ln^2x)$$
Since its $0\times\infty$ indeterminate form I tried to convert it into $...
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Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form
I came across this limit problem:
$\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$
Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
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Can someone explain when a limit is defined?
Why is it that the following limit is defined if $\lim_{x\to a}f(x) = \lim_{x\to a}g(x) = 0$?
$$\lim_{x\to a}\frac{f(x)}{g(x)}$$
In contrast, the limit isn't defined if $\lim_{x\to a}f(x) \neq 0$ but $...
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Can we say that $1^\infty$ is indeterminate because we can't classify $\infty$ as even or odd?
While studying indeterminate forms I like many others had used this post to understand why $1^\infty$ is indeterminate. Today I thought of a new and perhaps simpler argument to explain why this is so ...
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Evaluate $\lim\limits_{x\to\infty}x\!\left[2x\!-\!\left(x^3\!+\!x^2\!+\!x\right)^{\!\frac13}\!\!-\!\left(x^3\!-\!x^2\!+\!x\right)^{\!\frac13}\right]$ [closed]
Evaluate $\lim\limits_{x\to \infty}x\left[2x-\left(x^3+x^2+x\right)^{\frac{1}{3}}-\left(x^3-x^2+x\right)^{\frac{1}{3}}\right]$
My Approach:
Formula I used $(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^2+.....\...
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How to do this limit $L=\lim_{(x,y) \to (1,4)} \frac{y^2 - 4xy}{y^2 - 16x^2}$
Evaluate :
$$L=\lim_{(x,y) \to (1,4)} \frac{y^2 - 4xy}{y^2 - 16x^2}$$
My Work :
We can cancel $(y-4x)$ from the numerator and denominator provided $y \neq 4x$ and this comes out as $1/2$.
When the ...
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If the limit $\lim\limits_{x\to 0}{\frac{\sin 3x}{x^3} + \frac{a}{x^2}+b}$ exists and equals $0$ then what can $a$ and $b$ be?
Let $$L=\lim\limits_{x\to 0}{\frac{\sin 3x}{x^3} + \frac{a}{x^2}+b}=0$$
given that $a,b \in \mathbb R$ and are finite.
I tried the following approach,
We know, $\lim\limits_{x\to 0}{\frac{\sin 3x}{3x}}...
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How to compute the following limit? $\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$ [closed]
$$\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$$
Here, $\lfloor x \rfloor$ represents the floor of $x$.
I tried using a graphing calculator (desmos) to plot the function $...
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Limit evaluation, de l'Hopital seems not working
I have to evaluate
\begin{equation}
L=\lim_{x\to 0} \frac{1-e^{\frac{x^2}{2}}\cos x}{2\sin^2x -x \arctan2x}
\end{equation}
I tried with de l'Hopital's rule but it seems not working, am I missing ...
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Why is the graph of function behaving in a seemingly erratic fashion?
I was recently trying to plot the graph of the function $f(x)$. This function is defined as follows:
$
f(x) = \frac{\sin{3x} - 3\sin{x}}{(\pi - x)^3}
$
I first plotted the numerator ($A(x) = \sin{3x} -...
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How to get the limit of the sequence $(\sqrt{n^2+n}-\sqrt{n^2-1})$? [duplicate]
My Attempt
$$
\sqrt{n^2+n} - \sqrt{n^2-1}
= \sqrt{n+1} \, \bigl(\sqrt{n}-\sqrt{n-1} \bigr)
$$
Then I tried to apply the sandwich theorem in some way but failed.
Important Note
Please do not solve the ...
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Using L'Hospital's Rule for Fourier's series
The origin of this question comes from watching this YouTube video at 8:00.
The equation to evaluate is $$f(x)=\frac{1}{i(j-k)}e^{i(j-k)x}|_{-\pi}^{\pi}$$
However, this equation can be evaluated ...
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Does $ \lim_{x \to 0^+} (1 + x)^{\ln x}$ exist? [closed]
Does $\displaystyle \lim_{x \to 0^+} (1 + x)^{\ln x}$ exist?
I do not have any attempts solving it because I do not know how to transform it into the form in which I use l'Hôpital's rule.
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Can the following $\infty - \infty$ limit be solved without using l'Hôpital rule?
I know that some of the $\infty - \infty$ limits can be solved without using l'Hopital's rule. In such cases, you would usually be able to either rationalize (or derationalize if that is what its ...
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Bypassing the indeterminate form $0\cdot \infty$
Trying to calculate out the limit \begin{array}{rcl}
\lim_{x \to +\infty } (e^{x^{2}\sin \frac{1}{x}}-e^{x})&& \\
\end{array}
I come up with the indeterminate form $0\cdot \infty$ as ...
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How can we multiply÷ the expression by its conjugate in the $\infty-\infty$ indeterminate case? Is it not $\frac{\infty}{\infty}$?
Question
$$\lim _{x \rightarrow \infty} \left(x-\sqrt{x^2+5 x}\right)$$
To evaluate the limit, we multiply and divide the expression by its conjugate.
First Question
But since $x \rightarrow \infty$, ...
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$\infty-\infty$ indeterminate case
The methods yield two different answers. Could you explain the reason clearly and in detailed?
Question:
$$\lim_{{x \to \infty}} \left( \sqrt{x^2 + 6x + 14} - (x+1) \right) = ?$$
Solution:
Method 1:
...
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Indeterminate forms other than the 7 common ones
The following 7 indeterminate forms are all I can find in any calculus books:
$$\frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, \infty - \infty, 0^0, \infty^0, 1^\infty.$$
For example, by $\frac{0}...
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For smooth $f$ on $[0,1]$ such that $f^{(k)}(0)=0$ for all $k\in\mathbb{N}$, is it true that $\lim_{x\to0}\frac{xf'(x)}{f(x)}=\infty$?
Got asked this question and I got a bit surprised at how messy it became.
Suppose $f$ is a smooth function on $[0,1]$ such that the $k^{\text{th}}$ derivative $f^{(k)}(0)=0,\forall k\in \mathbb{N}$.
...
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Limit of a sequence: $\displaystyle \lim_{n\to\infty}\sqrt{n}\left[\frac{e^{\sqrt{n+1}}}{e^{\sqrt{n-1}}}-1\right]?$ [closed]
The sequence in question:
$$(u_{n})_{n\ge 0}=\sqrt{n}\left[\dfrac{e^{\sqrt{n+1}}}{e^{\sqrt{n-1}}}-1\right]$$
I see that there is an indeterminate form that must be lifted to calculate the limit, but I'...
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For which values of a this limit is zero?
I need to find the values of $a$ for which the following limit is $0$ :
$$L = \lim _{x\rightarrow 0^{+}}\frac{\sin(\log( 1+3x)) -e^{3x} +\cos x}{(\sin x)^{3\alpha }}$$
It is an indeterminate form of ...
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Proof of $\lim_{n\to \infty}(1 + \frac{b}{n})^n = e^b$ [duplicate]
I found one definition of $e$ to be $\lim_{n\to \infty}(1 + \frac{1}{n})^n = e$ and I checked it with a graphing calculator to be true. However the $1$ in the numerator is a special case, where it ...
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How do we evaluate $\lim_{x\to\infty} \left(\frac{x-c}{x+c}\right)^x$?
$\lim_{x\to\infty} (\frac{x-c}{x+c})^x=\frac{1}{4}$
My teacher did the following steps:
$\lim_{x\to\infty} \ln{(\frac{x-c}{x+c})^x}=\ln{\frac{1}{4}}$
$\lim_{x\to\infty} x\ln{\frac{x-c}{x+c}}=\ln{\frac{...
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Convergent values of series containing variables approache zero
Example 7 in Thomas' Calculus Early Transcendentals 14th edition p.614 demonstrates how to use Taylor series to find a limit involving an indeterminate form:
$$
\lim_{x\to 0}\left(\frac{1}{\sin(x)} - \...
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Electric Field Due to uniformly charged infinitely long wire
Calculate the field at a point P at distance r from infinitely long wire with charge density $\lambda$
Generally the electric field in this case at point P, distance r from the wire is derived using :...
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L'Hospital's Rule of infinity over infinity
I am troubled for understanding the L'Hospital's Rule of $\infty/\infty$ :
$$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}= \lim_{x\rightarrow a}\frac{f^\prime(x)}{g^\prime(x)} \tag{1}$$
where $\lim_{x\...
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Difference between 0+0+0+0...infinite times and 0 multiplied by infinity. [closed]
The measure/size of set I came across that the measure of a set that is formed by intersection of countably infinite disjoint sets of measure 0 each, will be the sum of the measures of all the sets = ...
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On the Divergence of an Improper Integral where the Integrand becomes unbounded in the Neighbourhood of a Point inside the Interval of Integration
I have been having some trouble with the way to determine the convergence/divergence of an improper integral such as this: $$\int_0^a \frac{\arctan{(x)}}{1-x^3}dx,$$ where $a>1.$ It is evident that ...
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3
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Tricky limit on the open first quadrant: $\lim_{(x,y) \rightarrow (0,0)} -x\log{y} $.
Consider $ f:(0,1) \times (0,1) \rightarrow \mathbb{R} $ where$$ f(x,y)=-x\log{y} $$ I am trying to prove whether $$ \lim_{(x,y) \rightarrow (0,0)}{f(x,y)}=0$$ My current idea is as follows:
Let $ p:(...
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Limits of indeterminate forms $(-\infty)(0)$
Can anybody give me a detailed solution on how
$$\lim_{x \to \infty} -x\left(1-e^{-\frac{1}{1+x}}\right) = -1?$$
I understand that separately the limit of each factor is $(-\infty)(0)$ but I could not ...
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0
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Limit of a indeterminate function is always can be calculated?
In class, a teacher made the following statement:
Whenever the limit of a real function of a real variable gives $\frac{0}{0}$ for $x \to a$, the limit can be calculated (via L'Hopital for example).
I ...
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2
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Advanced methods to explain indeterminate forms
I know this post may be covering a subject that is considered 'low quality' but I wanted to try and cover it in a more advanced manner (before writing I also searched if there were duplicate posts).
I ...
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Why $ 0^{\infty} $ isn't indeterminate form?
I was trying to calculate $$
\lim _{x \rightarrow 0} x^{\frac{1}{x}}
$$ I know left hand limit is not equal to right hand limit, hence limit doesn't exist. But I was trying to get their values as well....
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2
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Find the $\lim_{x\to 1} x^{\frac{1}{x^2-1}}$ without applying L'Hôpital's rule
Professor wants me to find the limit
$$
\lim_{x\to 1} x^{\frac{1}{x^2-1}}
$$
without using L'Hôpital and even gave the following advice:
Rewrite the exponent as a product of sum and difference, make ...
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answer
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Find Limit : $\lim_{x\to-∞}\frac {1}{xe^{x}}$
If I'm doing everything correctly, I get to 1/(-∞×0) where -∞×0 is undefined, so I don't know what to do. Also, I cannot do a series expansion because x is approaching infinity.
Can someone please ...
1
vote
1
answer
129
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Can indeterminate form be used as a critical point?
If we have a derivative, such as $(e^x-1)/x$, could we test the value of $x$ at $0$ to see it it’s a relative minimum or maximum (using the first derivative test) at that point? In other words, is it ...
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votes
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L'Hospital rule for $\frac{0}{0}\cdot\frac{a}{0}$ [closed]
Is there a solution to use the rule of L'Hospital for a boundary value problem like
$$\lim_{x\to1}\left(\dfrac{x^{2}-1}{x-1}\cdot\dfrac{a}{x^{3}-1}\right)$$
where $a> 0$ ?
I know how to solve $\...
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3
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How Do I Evaluate $\lim\limits_{n\rightarrow \infty} \sqrt[3]{(n+1)^2}-\sqrt[3]{(n-1)^2} $
I'm trying to evaluate this limit:
$$\lim\limits_{n\rightarrow \infty} \sqrt[3]{(n+1)^2}-\sqrt[3]{(n-1)^2}$$
It is an $\infty-$$\infty$ form. I've tried rewriting it as-
$$\lim\limits_{n\rightarrow \...
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1
answer
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How do you solve $\lim_{n\to\infty}\left(\frac{\log(n+1)}{\log(n)}\right)^{n\log(n)}$ [closed]
$\lim\limits_{n\to\infty}\left(\dfrac{\log(n+1)}{\log(n)}\right)^{n\log(n)}$
As it’s an indetermination, I’ve tried to do
$\exp\left(\lim\limits_{n\to\infty}(n\log(n))\cdot(\log(n+1)/\log(n)-1)\right)$...
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