Questions tagged [limits-without-lhopital]

The evaluation of limits without the usage of L'Hôpital's rule.

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1answer
18 views

How should I prove that $c^n/n!$, where c is a constant, tends to zero for a large n.

I want to prove that $$\frac{c^n}{n!}$$ tends to zero when $n$, a positive integer, is quite large. I do not think I can use L'Hospital's rule here as $n$ is discrete. Should we assume $n$ to be ...
2
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1answer
15 views

Find the limit $\lim_{x\to 0}\frac{\sqrt[5]{1+\sin x}-1}{\ln(1+\tan x)}$ [duplicate]

Find the limit $\lim_{x\to 0}\frac{\sqrt[5]{1+\sin x}-1}{\ln(1+\tan x)}$. I tried subtracting $1$ for using $\lim$ of $e$ but I realized that this is not a problem that can be solved in that way. ...
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2answers
132 views

Estimate limit without using L'Hopital

It is possible to estimate the following limit without using L'Hopital Rule? $$\lim_{x\to \frac{\pi }{3}}\left(\frac{\sin\left(x-\frac{\pi}{3}\right)}{1-2\cos\left(x\right)}\right)$$ I will be happy ...
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2answers
49 views

Calculate $\lim_{x \to a} \frac{\log(x-a)}{\log(e^x-e^a)}$ without using L'Hopital [duplicate]

I can calculate this easily using L'Hopital Rule. Can anyone give me some pointers on how to do this without using L'Hopital? $$\lim_{x \to a} \frac{\log(x-a)}{\log(e^x-e^a)}$$ I tried substitution ...
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0answers
33 views

How to prove this simple fact about limits: [closed]

If $$\lim_{c \to a^{+}}{f(c)}$$ exists, and $a<c<x$ then $$\lim_{x \to a^{+}}{f(c)}=\lim_{c \to a^{+}}{f(c)}$$ This doubt poped when reading L'Hôpitals rule proof for real limit. I am not going ...
3
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3answers
61 views

How do I evaluate the limit $\;\lim\limits_{x\to 0}\dfrac{\sin3x}{x\cos2x}$?

$$\lim_{x\to 0}\dfrac{\sin3x}{x\cos2x}$$ I'm having trouble doing this problem: the farthest I've gotten is just using a limit law for division and then moving the constant $x$ in the denominator in ...
3
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1answer
42 views

$\frac{1}{\cosh{x}}+\log\left ( \frac{\cosh{x}}{1+\cosh{x}} \right )$ for $x \rightarrow \pm \infty$ has a limit

Show from the definition of a limit that $$\frac{1}{\cosh{x}}+\log\left ( \frac{\cosh{x}}{1+\cosh{x}} \right )$$ for $x \rightarrow \pm \infty$ has a limit. My attempt This one is really tough for me. ...
2
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2answers
79 views

Find the limit of $\;\lim\limits_{x\to\infty}\left(\frac{x^2-1}{x^2+1}\right)^\frac{x-1}{x+1}$.

Find limit of $\;\lim\limits_{x\to\infty}\left(\dfrac{x^2-1}{x^2+1}\right)^\frac{x-1}{x+1}$ without using $L'Hopital$ I tried subtracting $1$ for using $lim$ of $e$ but I got $1^\infty$ form and ...
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5answers
95 views

Find $\lim_{x\to 0} \frac{1+\sin x-\cos x}{1+\sin(px)-\cos(px)}$

Find the limit of $$\lim_{x\to 0} \frac{1+\sin x-\cos x}{1+\sin(px)-\cos(px)}$$ without l'Hospital. I tried expanding $1-\cos(x)$ to get something this kind $$\lim_{x\to 0}\frac{\sin x}{x}$$ but ...
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1answer
45 views

Evaluating a limit using the Stolz-Cesàro theorem

I have been trying to compute the following limit- $$\lim_{n\to \infty} \dfrac {2021(1^{2020}+2^{2020}+3^{2020}....+n^{2020}) - n^{2021}}{2021(1^{2019}+2^{2019}+3^{2019}.....+n^{2019})} =L$$ By the ...
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1answer
35 views

Find limit (without using L' Hospital Rule) I can find this limit using L' Hospital Rule, I do not know how to do it without that [closed]

Find limit (don't use Lophital rule) $$\lim _{x\to 0}\left(\frac{\sqrt{1+x}\:-\sqrt{1-x}}{\sqrt[3]{1+x}-\sqrt[3]{1-x}\:}\right)$$ I can find this limit using L' Hospital Rule, I do not know how to do ...
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2answers
43 views

What is lim n→∞ (2)^(1/n)? Why is it equal to 1? [duplicate]

What is lim n→∞ (2)^(1/n)? Why is it equal to 1? All proofs shown are about n^(1/n) or something like this form. What about 2?
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0answers
20 views

Limit with a trigonometric function in the denominator

I had this simple looking limit: $$\lim_{x\to 0} \frac{3^{(2^{x+1}-2)}-1}{\sqrt{1-\cos(x)}}$$ By some algebraic manipulation, I got here: $$\frac{\sqrt{2}}{9}\lim_{x\to 0} \frac{3^{2^{x+1}}-9}{|\sin(x)...
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2answers
71 views

How do I find the limit of $\lim_{x \to 0} \frac{x-\ln(x+1)}{x(\sin(2x))}$?

How do I find the limit of $$\lim_{x \to 0} \frac{x-\ln(x+1)}{x(\sin(2x))}$$ I know it is equal to $\frac{1}{4}$, but how did we get that? Without using L'Hopital's rule. I tried canceling $x$ with $...
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1answer
47 views

Given a recursion $a_{n+1}=\sqrt{a_n^2+a_n}$ with $a_1=1.$ Prove that $\lim{\left(a_n\right)}'=\frac12$ without using $a_n\sim\frac n2-\frac14\ln n$

Given a recursion $a_{n+ 1}= \sqrt{a_{n}^{2}+ a_{n}}$ with $a_{1}= 1.$ Prove that $\lim{\left ( a_{n} \right )}'= \frac{1}{2}$ without using the result I've got $$a_{n}\sim\frac{n}{2}- \frac{1}{4}\ln ...
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1answer
43 views

Finding limit of [x]{x} when 'x' tends to 0.

I calculated $\lim_{x\to 0}$ [x]{x} as follows ({x} and [x] mean fractional part function and greatest integer function respectively):- $$\lim_{x\to 0} [x]\{x\}$$ To find the right hand limit, I ...
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2answers
165 views

computing the limit $\lim_{\theta \to \frac{\pi}{2}} (\sec \theta - \tan \theta)$

I'm trying to compute the following limit and would greatly appreciate your heartening feedback on my solution. The limit: $\lim_{\theta \to \frac{\pi}{2}} (\sec \theta - \tan \theta)$ My steps in ...
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1answer
98 views

A limit which exists in polar coordinates but not in Cartesian coordinates?

Let's have look at the function $$ f(x,y) \begin{cases} \frac{y(x^2+y^2)}{y^2+(x^2+y^2)^2} & (x,y)=(0,0) \\0 & (x,y)\neq(0,0)\end{cases}.$$ Switching to polar coordinates gives $$ f(r,\theta)=\...
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3answers
92 views

$\lim_{n\to\infty} \frac{a^{2n}+1}{a^n+2}.$

I would like to ask why is the sequence convergent to $\dfrac{1}{2}$ when the parameter $|a|<1$ ? At the same time, I would like to ask if there is a limit for $a<-1$, because the sequence would ...
2
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1answer
36 views

sequence limits - correctness of the adjustment

I would like to ask why the following modification of the example is not allowed $$\displaystyle\lim_{n\rightarrow \infty} \frac{1+2+ \cdots + 2^n}{1+5+ \cdots + 5^n} = \lim_{n\to \infty}\left(\frac{2^...
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3answers
79 views

$\lim_{n\to\infty} \frac{3^n}{(1+3)(1+3^2)(1+3^3)\ldots (1+3^n)}$

It is visible that the result is 0, but I can't calculate it. $\lim_{n\to\infty} \frac{3^n}{(1+3)(1+3^2)(1+3^3)\ldots (1+3^n)}$ It occurred to me to express it as a product and take a ratio test, but ...
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0answers
16 views

Finding Multivariable Limit w/ Conjugates

How do I find the multivariable limit of a conjugate? I know that I have to multiply the numerator and the denominator by the conjugate of the numerator so I can take out the square roots in the ...
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2answers
95 views

Formal proof for the limit of $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$

Formal proof for the limit of $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$. So far Keep in mind I have to use the definition for a limit. I.e for this would be a proof for the limit at $\...
3
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4answers
84 views

$\lim_{n\to\infty} \frac{2^n(n^4-1)}{4\cdot 3^n + n^7}$

How do I solve this example? I tried to point out the fastest growing term $2 ^ n$ and $3 ^ n$, but that doesn't seem to lead to the result. I know the limit is $0$ that's obvious, but I don't know ...
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1answer
26 views

Calculating a limit using Landau notation

I'm trying to do a problem but I can't figure out how to do it. We have to calculate a limit using Landau notation. I'm in the french schooling system so I'm not sure if this is what it is called but ...
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3answers
49 views

How to evaluate $\lim_{(x,y)\to (0,0)} \frac{x^4+y^4}{2x-y}$?

Find the limit : $$\lim_{(x,y)\to (0,0)} \frac{x^4+y^4}{2x-y}$$ I have tried using polar substitution but the problem is the denominator. Denominator can go to $0$ for some value of theta and is not ...
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1answer
56 views

How to compute $ \lim_{n\rightarrow\infty}{\frac{1+2\cdot2!+\dots+n\cdot n!}{(n+1)!}}$?

I'm asked to find the limit of the following sequence: $$ \lim_{n\rightarrow\infty}{\frac{1+2\cdot2!+\dots+n\cdot n!}{(n+1)!}}$$ I've tried using the Stolz Theorem with $a_n=1+2\cdot2!+\dots+n\cdot n!$...
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2answers
52 views

Proof limit value with $\epsilon/ \delta$

I have to find the limit value for $f(x)=\sqrt{1+x}$ for $x \rightarrow0$. And then show with $\epsilon /\delta$ that I have found the right limit value. I have found the limit value to $\sqrt{1+x} \...
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2answers
68 views

$\epsilon , \delta$-proof and choosing correct $\delta$

Let $$f(x)=\sqrt{1+x}$$ Show that if $x \rightarrow 0$ a limit value does exist. Furtheremore, find the limit value and explain the choice of $\delta$ w.r.t $\epsilon$ when the definitions of a limit ...
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3answers
58 views

Proof by induction: L'Hospital rule doesn't work, what next?

Show through induction: $\lim_{x\to\infty} \frac{\ln^k x}x=0$ Well the base case of 1 is rather trivial, but when substituting k for k+1 for the inductive hypothesis step we get, $\lim_{x\to\infty} \...
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3answers
85 views

Finding $\lim_{x\to-\infty}x\cdot e^x$ without using L'Hospital rule

I have to evaluate $$\lim_{x\to-\infty}x\cdot e^{x}$$ I know how I could do using L'Hospital rule, but I can't use it. I tried to rewrite it as: $$\lim_{x\to-\infty}\frac{x}{e^{-x}}$$ but I still get $...
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1answer
73 views

What is the limit of $f(x)=\frac{\sqrt{(x-2) \left(x^2+2 x-8\right)}}{x^2-4}$ when $x\to 0$?

I tried to find out the value of limit but when I took $x_-\rightarrow0$ then a result is obtained again when I took $x_+\rightarrow0$ another result was obtained. Can anyone please explain why is ...
1
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1answer
60 views

How do I solve this limit $\lim_{x\to \infty}\left(x-\sqrt{\frac{4x^3+3x^2}{4x-3}}\right)$?

I need to find the limit of $$\lim_{x\to \infty}\left(x-\sqrt{\frac{4x^3+3x^2}{4x-3}}\right)$$ I did rooting like this: $$\frac{\left(x-\sqrt{\frac{4x^3+3x^2}{4x-3}}\right)*\left(x+\sqrt{\frac{4x^3+3x^...
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2answers
37 views

Limit of $n^c/b^n$ when $n$ tends to infinity

How do I prove that $\lim_{n\to\infty}\frac{n^c}{b^n}=0$, where $c\in\mathbb{R}$ and $b>1$? I've managed to prove it using L'Hospital rule working with $\frac{x^c}{b^x}$, but is there a simpler way ...
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2answers
40 views

How to calculate the limit of the following function? [closed]

How can I calculate the limit of the following function? $$\lim_{(x,y)\to(0,0)}\frac{-|x+y|}{e^{x^2}{^{+2xy+y^2}}}$$
2
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3answers
52 views

l'Hospitals rule for multivariate limit

I have the following bivariate function where $\alpha,\beta,\gamma,\delta$ are constants, $a,b,c,d$ are integers and $x,y\in [0,1]$ $$ f(x,y) = \frac{\alpha(1-x^ay^b) + \beta(1-x^cy^d)}{\gamma(1-x^2)+\...
2
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2answers
73 views

Prove $\lim \frac{cos(x)-1}{x} = 0$ without l'Hopital [duplicate]

While deriving $\frac{\rm{d}}{\rm{dx}}\rm{sin}(x)$, using the definition of the derivative and expanding $\rm{sin}(x+h)$ leads to $\frac{\rm{d}}{\rm{dx}}\rm{sin}(x) = \rm{sin}(x)\lim_{h\to 0} \frac{...
2
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2answers
63 views

Solving limit without using l'hopitals rule

Find the limit as $h \to 0$ for $\frac{f(x)-f(x-h)}{h}$ at $x=0$, where $f(x) = x + |x|$ without using L'hopital rule. Once I plug in the numbers and try to work it down I get brought to $\frac{-(-h + ...
1
vote
1answer
43 views

$\lim_{x \to -\infty} \sqrt{4x^2-nx}+2x = \frac{n}{4}$

Prove that $$\lim_{x \to -\infty} \sqrt{4x^2-nx}+2x = \frac{n}{4}$$ My attempt: I simplified $\sqrt{4x^2-nx}+2x$ by conjugation $$\sqrt{4x^2-nx}+2x \left(\frac{\sqrt{4x^2-nx}-2x}{\sqrt{4x^2-nx}-2x}\...
3
votes
4answers
71 views

Limit of $\log(n!)/\log((n+1)!)$?

I want to show that $$\lim_{n\to\infty}\frac{\log(n!)}{\log((n+1)!)}=1.$$ Obviously, $$\frac{\log(n!)}{\log((n+1)!)}=\frac{\log(n!)}{\log(n+1)+\log(n!)}\leq1.$$ But I get stuck with the other estimate....
1
vote
3answers
67 views

Evaluate the limit $\lim_{n\to\infty}\sqrt{n} \int_{0}^{\frac{1}{n}}e^{-nx}\frac{x}{\sin x}dx$

Evaluate the limit$$\lim_{n\to\infty}\sqrt{n} \int_{0}^{\frac{1}{n}}e^{-nx}\frac{x}{\sin x}dx$$ I used L'Hopital's Rule $$\lim_{n\to\infty}\sqrt{n} \int_{0}^{\frac{1}{n}}e^{-nx}\frac{x}{\sin x}dx=\...
3
votes
5answers
120 views

Calculate $\displaystyle \lim_{x \to 3} \frac{\sqrt{19-x} - 2\sqrt[4]{13+x}}{\sqrt[3]{11-x} - x + 1}$

Calculate: $$\displaystyle \lim_{x \to 3} \frac{\sqrt{19-x} - 2\sqrt[4]{13+x}}{\sqrt[3]{11-x} - x + 1}$$ The problem with that case is that the roots are in different powers so multiplication in ...
0
votes
3answers
42 views

Calculate $\lim_{x \to 0^+} \frac{3x + \sqrt{x}}{\sqrt{1- e^{-2x}}}$

I need to calculate: $$\displaystyle \lim_{x \to 0^+} \frac{3x + \sqrt{x}}{\sqrt{1- e^{-2x}}}$$ I looks like I need to use common limit: $$\displaystyle \lim_{x \to 0} \frac{e^x-1}{x} = 1$$ So I take ...
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votes
2answers
35 views

Find the limit by L'hospital rule [closed]

Finding limit by l'hospital rule $$ \lim_{x \to 0} \frac{7^{2x}-5^{3x}}{2x - \arctan 3x} $$ Can anyone to explain me how the L'hospital rule works?
1
vote
1answer
35 views

Evaluating a limit at infinity

I was doing a physics problem when I had to compute an integral of the form $$I=\int\frac{dx}{\sqrt{a+x^2}} $$ for $a>0$, which is easily eavluated by making the substitution $x=\sqrt{a}\tan^{-1}(\...
0
votes
1answer
88 views

Calculate this limit without using L'Hôpital rule ..Any suggestions( I got to the solution with the suggestions, why do you close the questions?) [closed]

I can't think of any change of variable to solve this limit, I can't use derivatives or series. Can you please guide me on how to do it? Thank you $$\underset {x\to 6} {\text {lim}}\frac {x -\sqrt[3] {...
1
vote
1answer
37 views

calculating $\lim\limits _{x\to0}\left(\frac{1}{x^{2}}-\frac{e^{x}}{\left(e^{x}-1\right)^{2}}\right)$ using taylor polynomials

I am trying to calculate the limit $$\lim\limits _{x\to0}\left(\frac{1}{x^{2}}-\frac{e^{x}}{\left(e^{x}-1\right)^{2}}\right)$$ using taylor polynomials. I tried using L'Hôpital's rule, but it was ...
1
vote
2answers
87 views

$\displaystyle \lim_{x \to 0} \frac{\ln(1+\sin(2x^5))}{\tan^3(3x) \cdot (5^{x^{2}} - 1)}$ without using L'Hospital or Taylor's series

Find limit without using L'Hospital or Taylor's series: $$\displaystyle \lim_{x \to 0} \frac{\ln(1+\sin(2x^5))}{\tan^3(3x) \cdot (5^{x^{2}} - 1)}$$ $$\displaystyle \lim_{x \to 0} \frac{\ln(1+\sin(2x^...
0
votes
1answer
47 views

$\lim_{x\to 3}((x+1)(x-3)\tan(\frac{x\pi}{2}))$

Been wondering on how to compute this limit without: L'hôpital Any Taylor Series Only with trigonometric identities: $\lim_{x\to 3}((x+1)(x-3)\tan(\frac{x\pi}{2}))$ And yeah I've tried some stuff: $$...
1
vote
2answers
59 views

Evaluating $\lim_{x\to 0}\frac1{x^2}({\arctan(1+x^2)-\arcsin\frac{\cos x}{\sqrt{2}}})$

I've been struggling for a while with the following limit: $$L = \lim_{x\to 0}\frac{\arctan(1+x^2) - \arcsin\left(\frac{\cos x}{\sqrt{2}}\right)}{x^2}$$ I know that $L = 1$, since it seems obvious ...

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