Questions tagged [limits-without-lhopital]

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

The idea here is to evaluate the limit using standard limit theorems (algebra of limits, Sandwich/Squeeze Theorem, essentially without using any differentiation) and some standard limit formulas related to algebraic, trigonometric, exponential and logarithmic functions. Very often, Taylor series techniques prove fruitful in such problems as they allow for easy cancellation of powers and most terms evaluate to zero, leaving a simple expression for the limit.

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Proving the surprising limit: $\lim\limits_{n \to 0} \frac{x^{n}-y^{n}}{n}$ $=$ log$\frac{x}{y}$

A few months ago, while at school, my classmate asked me this curious question: What does $\frac{x^{n}-y^{n}}{n}$ tend to as $n$ tends to $0$? I thought for a few minutes, became impatient, and asked "What?" His reply, log$\frac{x}{y}$, was…
Maxis Jaisi
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Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?

The limit is $$\lim_{x\to0}\left(\frac{1}{\sin^2x}-\frac{1}{x^2}\right)$$ which I'm aware can be rearranged to obtain the indeterminate $\dfrac{0}{0}$, but in an attempt to avoid L'Hopital's rule (just for fun) I tried using the fact that $\sin…
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Very indeterminate form: $\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x \longrightarrow (\infty-\infty)^{\infty}$

Here is problem: $$\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x$$ The solution I presented in the picture below was made by a Mathematics Teacher I tried to solve this Limit without using derivative (L'hospital) and Big O…
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prove that $\lim\limits_{x\to 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}$

I'm trying to find a way to prove this: EDIT: without using LHopital theorem. $$\lim_{x\rightarrow 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}.$$ Honestly, I didn't come with any good idea. We know that $\lim_{x\rightarrow 1}x^{1/m}$ is $1$. I'd love…
user6163
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Finding the limit of $\sqrt[n]{{kn \choose n}}$

As part of homework I'm trying to find the limit of $\sqrt[n]{{kn \choose n}}$ (with $k\in\mathbb{N}$ a given parameter). I've seen This limit: $\lim_{n \rightarrow \infty} \sqrt [n] {nk \choose n}$., on which the only answer suggests using…
Nescio
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Find $\lim_{x \to \infty} \frac{\sin{x}+\cos{x}}{x}$

Find $$\lim_{x \to \infty} \Bigg(\frac{\sin{x}+\cos{x}}{x}\Bigg)$$ My thinking is $$-1 < \sin{x} < 1 $$ $$-1 < \cos{x} < 1.$$ Therefore $$-2 < \sin{x} + \cos{x} < 2$$ $$\frac{-2}{x} < \frac{\sin{x} + \cos{x}}{x} < \frac{2}{x}.$$ But is this…
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show $\lim_{x\to 0}\frac{e^x-1}{x}=1$ without L'Hopital

how would you show that $$\lim_{x\to 0}\frac{e^x-1}{x}=1$$ without using derivatives or l'hopital but using basic ideas that are generally introduced just before derivatives in a typical introductory calculus course. my attempt.. $$\lim_{x\to…
userX
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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 deriving the solution: Preliminary…
Meilton
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Solving limit without L'Hôpital

I need to solve this limit without L'Hôpital's rule. These questions always seem to have some algebraic trick which I just can't see this time. $$ \lim_{x\to0} \frac{5-\sqrt{x+25}}{x}$$ Could someone give me a hint as to what I need to do to the…
user145583
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Determine $\lim_{x \to 0}{\frac{x-\sin{x}}{x^3}}=\frac{1}{6}$, without L'Hospital or Taylor

How can I prove that $$\lim_{x \to 0}{\frac{x-\sin{x}}{x^3}}=\frac{1}{6}$$ without using L'Hospital or Taylor series? thanks :)
Iuli
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Relationship between l'Hospital's rule and the least upper bound property.

Statement of L'Hospital's Rule Let $F$ be an ordered field. L'Hospital's Rule. Let $f$ and $g$ be $F$-valued functions defined on an open interval $I$ in $F$. Let $c$ be an endpoint of $I$. Note $c$ may be a finite number or one of the symbols…
RitterSport
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Prove limit converges in definition of $e.$

I've looked up several related questions, but they do not answer what I am looking for. Please give link if this is a duplicate. What I eventually want to know is why $$\lim_{n\to\infty}\left(1+\frac1n\right)^n$$ converges to a finite value. I…
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Long limit question

If $$\lim_{x \to \infty}\frac{a(2x^3-x^2)+b(x^3+5x^2-1)-c(3x^3+x^2)}{a(5x^4-x)-bx^4+c(4x^4+1)+2x^2+5x}=1$$ then find the value of $a+b+c$. I have done such problems with $x$ tending to some finite amount. Infinity is creating some problems.
user167045
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Without superior math, can we evaluate this limit?

We all knew, with $$\lim_{x\to 0}\frac{\sin x - x}{x(1-\cos x)}$$ we can use L'Hôpital's rule or Taylor series to eliminate undefined form. But without all tools, by only using high school knowledge, how can we evaluate this limit? It seems…
MathChill
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Trigonometry limit's proof: $\lim_{x\to0}\frac{\sin(x)+\sin(2x)+\cdots+\sin(kx)}{x}=\frac{k(k+1)}{2}$

How to prove that $$\lim_{x\to0}\frac{\sin(x)+\sin(2x)+\cdots+\sin(kx)}{x}=\frac{k(k+1)}{2}$$ I tried to split up the fraction and multiple-divide every new fraction with its $x$ factor but didn't work out. ex: $$\lim_{x\to 0}\frac{\sin(2x)}{x} =…
Leos Kotrop
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