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.

2796 questions
1
vote
4 answers

Computing: $\lim\limits_{x\to 0} \frac{(\cos x-1)(\cos x-e^x)}{x^n}$

$$\lim_{x\to 0} \frac{(\cos x-1)(\cos x-e^x)}{x^n}$$ If the limit is a finite non zero number, find n. I tried using L'Hôpital's rule but failed. I differentiated it but not understanding what exactly should be done. Please help.
1
vote
2 answers

Proving that $\begin{equation*} \lim_{x \rightarrow 1} |x + 2|=3 \end{equation*}$ using limit definition.

Prove that $\begin{equation*} \lim_{x \rightarrow 1} |x + 2|=3 \end{equation*}$ using limit definition> I have a problem when I substitute for $|x + 2|$ by $-x-2$, could anyone help me in this case please? I could not reach $|x-1|$ which is less…
user426277
1
vote
3 answers

Find limit $\lim_{x\to 1} \dfrac{4x^2\sqrt{x+3}-17x+9}{(x-1)^2}$

Find the following limit without using L'Hopital's rule: $$\lim_{x\to 1} \dfrac{4x^2\sqrt{x+3}-17x+9}{(x-1)^2}$$ My attempt: $$x-1=u\implies x=u+1$$ So we have: $$\lim_{u\to 0} \dfrac{4(u+1)^2\sqrt{u+4}-17(u+1)+9}{(u)^2}$$ What now?
Almot1960
  • 4,736
  • 13
  • 32
1
vote
5 answers

Proof $\lim\limits_{n \rightarrow \infty} \sqrt{n} (3^{\frac{1}{n}} - 2^{\frac{1}{n}}) = 0$

How would one go about proving the limit of sequence: $$\lim_{n \rightarrow \infty}\ \sqrt{n} (3^{\frac{1}{n}} - 2^{\frac{1}{n}}) = 0\,. $$ Epsilon definiton of limit seems to be too complicated and/or unsolvable (variable both in exponent and…
1
vote
3 answers

Limit of $f(x)$ when $x$ goes to zero

Let $f(x) = \frac{1 + \tan x + \sin x - \cos x}{\sin^2 x + x^3}$ . Find value of $\lim_{x \to 0} f(x)$ if it exists . I can solve it using L'Hospital's Rule and Taylor series but I'm looking for another way suing trigonometric identities .
S.H.W
  • 4,251
  • 4
  • 18
  • 46
1
vote
7 answers

How to solve $\lim_{x\to0^+} \frac{\sqrt{\sin x}}{\sin\sqrt{x}}$ without the use of L'Hospital's Rule

I have tried manipulating the expression to get rid of the indeterminate form. I tried to use intervals and the sandwich theorem.
1
vote
4 answers

Limit $\lim\limits_{x \to π/2}\frac{2x\sin(x) - π}{\cos x}$ without l'hospital

There is a problem that is easy to solve with L'hospital but we are required to solve it without it, but I could not find the answer. x* sinx part is especially confusing me, because other examples I have solved did not include such part. $$\lim_{x…
Rockybilly
  • 195
  • 7
1
vote
2 answers

How can I calculate the following difficult limit: $\lim_{x\to\infty} (x + \sin x)\sin \frac1x$?

How can I calculate the following limit? $$\begin{equation*} \lim_{x \rightarrow \infty} (x + \sin x)\sin \frac{1}{x} \end{equation*},$$ First, I multiplied and then distributed the limit then the limit of the first term was 1 but the second term…
Emptymind
  • 1,727
  • 17
  • 44
1
vote
2 answers

A difficulty in using squeeze theorem.

Given that: $\forall x \in \mathbb{R}, \sin^2(3x) \leq x^2f(x)\leq 3x^2 + 6\sin(x^2),$ then $$ \begin{equation*} \lim_{x \rightarrow 0} f(x) = \end{equation*}$$ i)0 ii)9 iii)$\infty$ iv) The limit does not exist. My idea is to divide by $x^2$ the…
Emptymind
  • 1,727
  • 17
  • 44
1
vote
1 answer

$\lim_{x\rightarrow 0} \frac{\sin(\sin x)}{x}$

Can someone suggest how to solve this limit? $$\lim_{x\to 0}\frac{\sin(\sin x)}{x}$$ If I substitute $y=\sin x$ then $\sin(\sin x)=\sin y$ while $x=\arcsin y$. Then the limit becomes $$\lim_{y\to 0}\frac{\sin y}{\arcsin y}$$ but this form is more…
Anne
  • 2,833
  • 10
  • 25
1
vote
2 answers

The limit of $1/x$ as $x\to 0$.

I need help with the following limit. I know that the answer is positive infinity but could someone break this down to me step by step please. I got confused since I intuitionally tried to plug in $x=0$ but then we can't divide by zero! Also we did…
user443248
1
vote
2 answers

Limit of an Exponential Equation

My problem is the $\lim_{x \rightarrow \infty} \left(1-\frac{4}{x}\right)^{3x}$. I got $$\ln(B) = \ln\left(\lim_{x \rightarrow \infty} 3x \cdot \ln\left(1-\frac{4}{x}\right)\right)$$ which equates to…
1
vote
3 answers

Calculating $\lim_{x\to0} \frac{tan^4(2x)}{4x^4}$

I’m trying to solve the following limit $$\lim_{x\to0} \frac{\tan^4(2x)}{4x^4}$$ I have made this process. $$\lim_{x\to0} \frac{\frac{1}{2}\tan^4(2x)}{\frac{1}{2}4x^4}$$ then I use the identity $$ \lim_{x\to 0} \frac{\tan(x)}{x}=1$$ and the…
Karl Jimbo
  • 57
  • 1
  • 7
1
vote
2 answers

How can I evaluate $\lim_{n \rightarrow \infty} \frac{n\sin n}{2n^2 - 1}$?

How can I evaluate $$ \lim_{n \rightarrow \infty} \frac{n\sin n}{2n^2- 1}? $$ Unsuccessful attempt: In the expression $\frac{n\sin n}{2n^2 - 1}$, I divided the numerator and denominator by $n^2$, but I got stuck with $\frac{\sin n}{n}$ and I do…
1
vote
3 answers

Determining the limit of a function containing $(x-1)^5$ without l'hopital`s rule.

Find the limit of $$\begin{equation*} \lim_{x \rightarrow 0} \frac{(x-1)^5 + (1 + 5x)}{x^2 + x^5} \end{equation*}$$ Shall I use the binomial theorem? Any hint will be appreciated!
user426277
1 2 3
99
100