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Questions tagged [limits-without-lhopital]

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

2
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2answers
24 views

Finding the limit of $2^{-1/\sqrt {n}}$ as $n \rightarrow \infty$.

Finding the limit of $2^{-1/\sqrt {n}}$ as $n \rightarrow \infty$. I feel like I can put $n = m^2$, them my limit will be $2^{-1/m}$ as $n \rightarrow \infty$. which is equal to lim $\frac{1}{2^{1/m}}$...
1
vote
2answers
47 views

Find the limit of $(1^7 + 2^7 + …+ n^7)^{1/n}$ as $n \rightarrow \infty$

The question tells me find the limit of $(1^7 + 2^7 + .......+ n^7)^{1/n}$. I thought that I would use an idea similar to the one given below: but with $b = n^{14/n}$ and using the $14^{th}$ ...
5
votes
4answers
102 views

Compute $\lim_{x\to0}{1 - e^{-x}\over e^x - 1}$

Compute $\displaystyle \lim_{x\to0}{1 - e^{-x}\over e^x - 1}$. So first and foremost, I know what the answer to this question is: $$\begin{align}\lim_{x\to0}{1 - e^{-x}\over e^x - 1} &= \lim_{x\...
0
votes
3answers
41 views

Evaluating $\lim_{x\rightarrow\infty}\left(\frac{2x-1}{3x+2}\right)^x$.

I have been trying to solve this limit but i think it doesnt get me anywhere. I tried with ln(y) but nothing. I tried to transform it to inf/inf but no result . Can anyone please help me find it ...
1
vote
2answers
39 views

Basic calc - Finding the extended function

I am not sure why but I'm having a really tough time with the following problem: Given: $$f(x) = \frac{11^x - 1}{x}$$ what should the extended function's value(s) be so that the function is ...
4
votes
3answers
97 views

Finding the limit by using the definition of derivative.

Here is the problem. Let $f$ be the function that has the value of $f(1)=1$ and $f'(1)=2$. Find the value of $$ L = \lim_{x \to 1} {\frac{\arctan{\sqrt{f(x)}-\arctan{f(x)}}}{ \left (\arcsin{\sqrt{...
1
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1answer
47 views

Evaluating limit in terms of alpha

In a calculus book I found: Evaluate the following limits in terms of the number $\alpha =\lim_{x \to 0} \frac{\sin x}{x}$ i) $$ \lim_{x \to 0} \frac{\sin 2x}{x} $$ I know some techniques how ...
0
votes
1answer
26 views

Proving a limit is equal to the derivative of function

Let $p(x)$ be a polynomial and consider the limit $\lim_{h\to0} \frac{p(x+3h)+p(x-3h)-2p(x)}{h^2}$ I am supposed to prove that it is equal to $9p''(x)$ which is simple using l'hopital rules. However,...
2
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8answers
79 views

find the limit of $\lim_{x \to\infty} (\frac {2+3x}{2x+1})^{x+1}$ without L'Hopital

$$ \lim_{x\to\infty}\left(\frac {2+3x}{2x+1}\right)^{x+1} $$ Not sure how to deal with this , I've tried doing the following $$ \lim_{x\to\infty}\left(\frac {2+3x}{2x+1}\right)^{x} . \lim_{x\to\...
2
votes
3answers
68 views

If $x_n\to 1$ can we compute this indeterminate limit?

If we let $x_n\to1$ and $y_n\to1$ can we understand the convergence of $$\dfrac{1-2x_n + x_n^2y_n}{(1-x_n)^2}?$$ This is a problem I've been thinking about for a while but can't actually pin down a ...
2
votes
3answers
84 views

Find the limit without using L'Hopital's rule $\lim_{x \to 0} (x-\tan x)/(x \tan x)$.

I solved it with L'Hopital's rule but I want to find out how can I solve it without using L'Hopital's rule.
1
vote
4answers
81 views

$\lim_{x \to 2} \frac{\cos{\left(\frac{\pi}{x}\right)}}{x-2}$ without using De L'Hospital [duplicate]

$$\lim_{x \to 2} \frac{\cos{\left(\frac{\pi}{x}\right)}}{x-2}$$ This limit is supposed be found without L'Hospital's Rule, but I have not been able to get close to the answer using conjugates, ...
4
votes
6answers
104 views

How to find $\lim_{x \to 0}\frac{1-\cos(2x)}{\sin^2{(3x)}}$ without L'Hopital's Rule.

How would you find $\displaystyle\lim_{x \to 0}\frac{1-\cos(2x)}{\sin^2{(3x)}}$ without L'Hopital's Rule? The way the problem is set up, it makes me think I would try and use the fact that $\...
0
votes
1answer
27 views

On Finding The Derivative At Asymptotes

I have 2 questions here. First. Why can't we find the derivative of a function at the vertical asymptote? Second. Can we find the derivative a function at it's horizontal asymptote, and if so/not, ...
0
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3answers
39 views

Evaluating the limit of $x^2 (1-\cos\frac{1}{x})$ when $x$ approaches infinity

I wanted to evaluate the limit $$\lim_{x\to\infty}x^2(1-\cos\frac{1}{x})$$ Since we know that $-1\leq \cos x\leq1$ and that $-1\leq \cos\frac{1}{x} \leq 1$, so by algebraic manipulation, $0\leq x^2(...
2
votes
1answer
49 views

Meaning Of $\lim_{x\to c}f(c)$

I'm coming across a problem that states $\lim_{x\to c}f(c)=f(c)$. But I'm really confused by this. What does $\lim_{x\to c}f(c)=f(c)$ even mean? For instance let's say $f(x)=x^2$, and $c=2$. So $\...
4
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6answers
105 views

How can I show $\lim_{x\to 0}\frac{e^{3x}-e^x}{x}=2$ without derivatives?

So I was looking through a Calculus book, in their first chapter section 2 about limits and I came across the following problem: $$ \lim_{x\to 0} \dfrac{e^{3x}-e^x}{x}=2 $$ Immediately, I took the ...
0
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1answer
44 views

Calculate $\lim_{x\to 0}\frac{e^{\frac{\ln{(1+x)}}{x}}-e}{x}$

Calculate $$\lim_{x\to 0}\frac{e^{\frac{\ln{(1+x)}}{x}}-e}{x}$$ My Attempt: $$\lim_{x\to 0}e\cdot \frac{e^{\frac{\ln{(1+x)}}{x}-1}-1}{\frac{\ln{(1+x)}}{x}-1} \cdot \frac{\frac{\ln{(1+x)}}{x}-1}...
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votes
3answers
77 views

Calculate $\lim_{x\to{0^+}}\frac{\log_{\sin{x}}{\cos{x}}}{\log_{\sin{\frac{x}{2}}}\cos{\frac{x}{2}}}$

Calculate $$\lim_{x\to{0^+}}\frac{\log_{\sin{x}}{\cos{x}}}{\log_{\sin{\frac{x}{2}}}\cos{\frac{x}{2}}}$$ My Attempt: $$\lim_{x\to{0^+}}\frac{\ln{(1-2\sin^2{\frac{x}{2}})}}{\ln{(1-2\sin^2{\frac{x}{4}...
1
vote
3answers
51 views

On "Proving' Infinite Limits, And A Counterexample

The textbook I am using has come up with this method to prove why in infinite limits, the term with the highest degree will dominate: It then goes on to say that at infinity, 1000/3x and all the ...
6
votes
6answers
2k views

On Infinite Limits

I am currently learning about infinite limits in Calculus, basically determining the limit of a function as x approaches infinity. However, I am struggling to understand the method being used to find ...
1
vote
3answers
91 views

Limit of $\lim\limits_{x \rightarrow \infty}{(x-\sqrt \frac{x^3+x}{x+1})}$ - calculation correct?

I just want to know if this way of getting the solution is correct. We calculate $\lim\limits_{x \rightarrow \infty} (x-\sqrt \frac{x^3+x}{x+1}) = \frac {1}{2}$. \begin{align} & \left(x-\sqrt \...
1
vote
4answers
54 views

Understanding The Squeeze Theorem

I'm having trouble understanding the squeeze theorem, or why, given $f(x)\leq g(x)\leq h(x)$, if the limit of $f(x)$ is $L$ and the limit of $h(x)$ is $L$, the limit of $g(x)$ will also be $L$. I ...
6
votes
1answer
102 views

If $f(x) =\frac{1}{3} ( \frac {5}{f(x+2)}+f(x+1))$ then $\underset{x \to \infty}{\lim}f(x) = ?$

If $f(x) =\frac{1}{3} ( \frac {5}{f(x+2)}+f(x+1))$ and $f(x) > 0$ for all $x \in \mathbb R$ then $\underset{x \to \infty}{\lim}f(x) = ?$ can anyone please give me a hint to find it? I know how ...
1
vote
2answers
59 views

Evaluate $\lim _{x \to 0} [2x^{-3}(\sin^{-1}x - \tan^{-1}x )]^{2x^{-2}}$=? [duplicate]

$$\lim _{x \to 0} \left[\frac{2}{x^3}(\sin^{-1}x - \tan^{-1}x )\right]^{\frac{2}{x^2}}$$ How to find this limit? My Try: I tried to evaluate this $$\lim _{x \to 0} \left[\frac{2}{x^3}(\sin^{-1}x - \...
1
vote
6answers
54 views

Find $\lim_{x \to 0 }\frac{(1+x)^{(1/2)} -1}{(1+x)^{(1/3)} -1}$

Find the limit without using L'hopital $$\lim_{x \to 0 }\frac{(1+x)^{(1/2)} -1}{(1+x)^{(1/3)} -1}$$ I have tried multiply by the conjugate of the denominator and numerator but didn't work Any ...
0
votes
4answers
70 views

Proof by $\epsilon-\delta$ definition $\lim_{x \to 0} \frac{e^x \cos x - 1}{x} = 1$

Proof by definition $\lim\limits_{x \to 0} \dfrac{e^x \cos x - 1}{x}=1$. I could not go far because I am lacking insight of how to find $\delta>0$ in this case. Let $\epsilon>0$. $\...
2
votes
2answers
78 views

Evaluate $ \lim _{x \to 0} \left[{\frac{x^2}{\sin x \tan x}} \right]$ where $[\cdot]$ denotes the greatest integer function. [duplicate]

Evaluate $$\lim _{x \to 0} \left[{\frac{x^2}{\sin x \tan x}} \right]$$ where $[\cdot]$ denotes the greatest integer function. Can anyone give me a hint to proceed? I know that $$\frac {\sin x}{x} &...
1
vote
4answers
70 views

Find the limit of $\lim_{ x\to 0 } \sin^{-1}(x) /\sin(3x)$ without using L'Hopital's rule [closed]

I'm new to calculus and I'm not sure how to deal with the inverse $\sin$ here. $$ \lim_{x \to 0} \frac{ \sin^{-1}(x) }{\sin(3x)} $$
1
vote
4answers
94 views

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$

Evaluate $\lim_{n \to \infty} ((15)^n +([(1+0.0001)^{10000}])^n)^{\frac{1}{n}}$ Here [.] denotes the greatest integer function. My Try : I know how to solve this kind of problem :$\lim_{n \to \infty}...
1
vote
3answers
135 views

How do you solve $\lim_{n\to \infty}{n[(1+\frac{c}{n})^{n}-(1-\frac{c}{n})^{-n}]}$ without L’Hopital's Rule? [closed]

I know that solution is: $-c^2e^c$ $$\lim\nolimits_{n\to \infty}{n\left[\left(1+\frac{c}{n}\right)^{n}-\left(1-\frac{c}{n}\right)^{-n}\right]}$$ Hint: common factor to $\left(1+\frac{c}{n}\right)^{...
0
votes
2answers
61 views

Find, from first principles, the derivative of $\log(\sec(x^2))$

Find from first principles, the derivative of $\log (\sec (x^2))$ My Attempt : Let, $y=f(x)=\log (\sec (x^2))$ $f(x+h)=\log (\sec (x+h)^2)$, where $h$ is a small increment in $x$ By first principle,...
1
vote
3answers
64 views

Prove $\lim_{(x,y) \to (0,0)} \frac{\exp(xy)\cdot xy\cdot(x^2-y^2)}{x^2+y^2} =0$.

I want to show that $$\lim_{(x,y) \to (0,0)} \frac{\exp(xy)\cdot xy\cdot(x^2-y^2)}{x^2+y^2} =0.$$ Is it valid to do it like this: $$\lim_{(x,y) \to (0,0)} \left|\frac{\exp(xy)\cdot xy\cdot(x^2-y^...
1
vote
3answers
66 views

Does the limit rule $\lim_{x \rightarrow 0}\frac{\sin x}{x}=1$ apply to $\lim_{x \rightarrow \pi}\frac{\sin\left(x-\pi\right)}{\left(x-\pi\right)}=1$?

In my textbook, I was given an example below : $$\lim_{x \rightarrow \pi}\frac{\sin\left(x-\pi\right)}{\left(x-\pi\right)}=1$$ Previously I was taught that this formula : $$\lim_{x \rightarrow 0}\...
1
vote
4answers
50 views

How to deal with $g(f(x))$ in the limit?

Let $\lim\limits_{x \to \infty} f(x)=\infty$. Does the following hold as $x\to\infty$?: $$\exp(x) \leq \exp\left({\frac{x}{1+1/\ln(1+f(x))}}\right)$$ My effort: If above inequality holds in the limit,...
0
votes
3answers
42 views

Question regarding limits of multivariable functions - is it really zero?

I am given the following problem in a textbook, and it's a solved problem. Find $\displaystyle \lim_{(x,y) \to (0,0)} \frac{x^2 y^2}{x^4 + 3y^4}$ or justify its nonexistence. The author justifies ...
0
votes
4answers
89 views

How to determine $\lim_{h \to 0} \frac{1- 9^h}{h}$ [closed]

How to determine $\lim_{h \to 0} \frac{1- 9^h}{h}$. Is it $3\log 3$? Actually my question is a sub part of another question in which I have solved everything but this.
0
votes
2answers
64 views

Solving limit $ \lim_{x\rightarrow 1}\left(\frac{\sqrt[3]{7+x^3}-\sqrt[2]{3+x^2}}{x-1}\right) $ without L'Hopital.

I was trying all the day to resolve this problem with different method without L'Hopital but I can't do it, I would really like to up my mathematical development but the post doesn't allow me because ...
0
votes
1answer
40 views

Sequences that diverge, limit stuff

I have a quadratic based sequence, say its of the general form : $y = ax^2 + bx +c$ To put a concrete example the sequence : $a(n) = n^2 +5n +6$ $\lim{a(n)}\to\infty$ when $n\to\infty$ I am ...
1
vote
5answers
85 views

Find the limit $\lim_{x\to0}\frac{\tan6x)}{\sin3x}$ without using L'hopital

I'm trying to compute the following limit: $$\lim_{x\to0}\frac{\tan6x}{\sin3x}$$ I really have no idea how to start it. I tried rewriting $\tan6x$ in terms of $\sin6x$ and $\cos6x$ but wasn't able to ...
1
vote
4answers
67 views

What is the next limit equal to?

$$\lim_{n\to\infty}(\sqrt{n^2+n}-\sqrt[3]{n^3+1})=?$$ I tried amplifying the hole substraction to form the formula $$a^3-b^3$$ but didn't worked out. Can you help me figure it out?
1
vote
1answer
50 views

Why does changing the operator in $\lim_{h\to0}$ alter the result of this function?

Let $f(x) = |x|$. Attempting to differentiate $f(x)$ at 0 will fail because the limit does not exist at 0 as the left and right side are unequal. $$\lim_{h\to0}\dfrac{f(0+h)-f(0)}{h}$$ However, my ...
0
votes
1answer
68 views

How to evaluate $\lim_{x\to+\infty}\frac{\sin\sin4x}{5x}$?

I'd like to learn how to evaluate this limit: $$\lim_{x\to+\infty}\frac{\sin\sin4x}{5x}$$ I tried to substitute with a new variable: Let $u=\sin4x$. Then as $x\to+\infty$, $u\to\,??$ Since $\sin x$...
1
vote
2answers
40 views

Appropriate change of variable for limit (sine function)

I couldn't find this particular solution on here; I apologise in advance if it has been posted before. I know that $\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$ I am asked to find this limit, using ...
1
vote
1answer
27 views

Limit of a weighted sum raised to a power

My professor in class used the below result to generalise the concept of averages- $$\lim_{n\rightarrow0}(\sum_{j=1}^{m}w_ja_j^n)^{1/n}=\prod_{j=1}^ma_j^{w_j}$$ $$\sum_{j=1}^{m}w_j=1$$ All the ...
1
vote
4answers
103 views

Is $\lim_{x\to -\infty} \frac{5x+9}{3x+2-\sqrt{4x^2-7}}=1$?

Evaluate $$\lim_{x\to-\infty} \frac{5x+9}{3x+2-\sqrt{4x^2-7}}$$ My attempt: $$\lim_{x\to -\infty} \frac{5x+9}{3x+2-\sqrt{4x^2-7}}=\lim_{x\to-\infty}\frac{5x}{-\sqrt{4x^2}}=\frac{-5}{2}$$ According to ...
2
votes
1answer
55 views

Isoceles triangle inscribed inside circle

QUESTION ABC is an isoceles triangle inscribed in a circle of radius $r$.If AB=AC and $h$ is the altitude from A to BC and $p$ be the perimiter of ABC then $$\lim_{h\to0} \frac{\Delta}{p^3}$$ (where $...
2
votes
2answers
47 views

What can we say about the series $\sum_{n=1}^{\infty}n\cdot (1-\cos(\frac{a}{n}))$

Series $\sum_{n=1}^{\infty}n\cdot (1-\cos(\frac{a}{n}))$ $$\sum_{n=1}^{\infty}n\cdot(1-\cos(\frac{a}{n}))=\sum_{n=1}^{\infty}n\cdot[1- \{1-\frac{a^2}{2!\cdot n^2} + O(\frac{1}{n^4})\}] = \sum_{n=1}^{\...
1
vote
1answer
32 views

Question about the rules to follow when evaluating infinite limits

I learned to solve infinite limits diving the whole function by the leading term, knowing that when you have a fraction, if it is in the numerator you will get $\infty $, and if it is in the ...
1
vote
5answers
58 views

How to evaluate $\lim _{x\to 0}\frac{5-5\cos\left(2x\right)+\sin\left(4x\right)}{x}$ without using L'Hospital's rule?

I need to evaluate the following limit without using L'Hopital's rule: $$\lim _{x\to 0}\left(\frac{5-5\cos\left(2x\right)+\sin\left(4x\right)}{x}\right)$$ I thought the best way was to separate it ...