Linked Questions

1
vote
4answers
22k views

Limit of sin(1/n)*n [duplicate]

My Maple input limit(sin(1/n)*n,n=infinity); says 1. I don't understand why $$ \lim_{n \to \infty} \sin\left(\frac{1}{n}\right) \cdot n = 1 $$ I know that $\lim_{...
1
vote
4answers
1k views

Limits of cosine and sine [duplicate]

When $\theta$ is very small why $\sin \theta$ is similar to $\theta$ and $\cos\theta$ similar to $1$? Is it related to limits or we can prove it simply by using diagrams?
1
vote
2answers
11k views

Why the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0 is 1? [duplicate]

I need a rigorous proof that verify why the limit of $\dfrac{\sin(x)}{x}$ as $x$ approaches $0$ is $1$. I tried before but i do not know how start this proof. I would appreciate if somebody help me. ...
0
votes
2answers
2k views

Is $\frac{\sin(x)}{x}$ continuous at $x=0$? Whats the value at $x=0$? [duplicate]

Is $\dfrac{\sin(x)}{x}$ at $x = 0$ continuous? Whats the value at $x=0~?$
2
votes
2answers
527 views

How to show that $\frac{\sin(n)}{n}$ is $1$ as $n \rightarrow 0$? [duplicate]

Possible Duplicate: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How to show that $\frac{\sin(n)}{n}$ is $1$ as $n \rightarrow 0$? just hint.
1
vote
4answers
348 views

About $ \lim_{x\rightarrow 0}\frac {\sin x}{x} = 1$ [duplicate]

I do not understand how $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ As if $$ x = 0, \frac{\sin (0)}{0} = \frac {0}{0} $$ So if someone could explain this I would appreciate it! Thanks!
0
votes
0answers
3k views

How do I prove that the limit as x approaches 0 of (sin(x)/x) = 1 using the definition of a limit? [duplicate]

If I let r be a positive number, I have to show that there exists a positive number, u, such that |x| < u implies that |(sin(x))/x - 1| < r. Now I've gone about this two different ways, but hit ...
2
votes
4answers
324 views

Why does $\lim_{x\rightarrow 0}\frac{\sin(x)}x=1$? [duplicate]

I am learning about the derivative function of $\frac{d}{dx}[\sin(x)] = \cos(x)$. The proof stated: From $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$... I realized I don't know why, so I wanted to learn ...
0
votes
4answers
882 views

How lim $h->0$ $\sin h/h$ is equal to $1$? [duplicate]

How $\lim_{h\to 0} \sin h$ when divided by $h$ gives the value $1$ ? Does it also follows with other ratios that for example. $\lim_{h\to 0} \cos x/x = 1$
1
vote
4answers
184 views

How is $\frac{\sin(x)}{x} = 1$ at $x = 0$ [duplicate]

I have a function: $$\text{sinc}(x) = \frac{\sin(x)}{x}$$ and the example says that: $\text{sinc}(0) = 1$, How is it true? I know that $\lim\limits_{x \to 0} \frac{\sin(x)}{x} = 1$, But the graph of ...
1
vote
3answers
192 views

$\lim_{x \to 0}\frac{\sin x}{x}$ intuition [duplicate]

I am looking for some basic intuition for $$\lim_{x \to 0}\frac{\sin x}{x}$$ If we look at it as $f(x)=\sin x\cdot\frac{1}{x}$ so as ${x \to 0}$ $\sin x \to 1$ and $\frac{1}{x} \to \infty $ so it is $...
2
votes
4answers
146 views

Limit $\lim_{x \to 0 }\frac{x}{\sin x} = 1$? [duplicate]

I have a question regarding limits. Recently in a math class, my teacher states that $\frac{\sin x}{x}$ goes to $1$ hence in the case of a $\lim_{x\to 0} \frac{x}{\sin x}$, the answer is $1$. Why is ...
-1
votes
2answers
153 views

Why is tangent of x greater than x? [duplicate]

I've seen the proof of $\lim_{x\rightarrow0} \frac{\sin(x)}{x}=1$ but it uses the fact that $\tan(x)>x$ which in all places says it is obvious but this is maths and a I'd like to see a purely ...
0
votes
1answer
140 views

Finding $\lim_{x\to 0}\frac{\sin x}{x} $ [duplicate]

How to find $$\lim_{x\to 0}\frac{\sin x}{x} $$
0
votes
1answer
92 views

Why does $\lim_{x\to0}\frac{\sin(x)}{x}$ equal $1$? [duplicate]

Without using L'Hospital's Rule, why does $$\lim_{x\to 0} \frac{\sin x}{x} = 1?$$

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