If using L'Hopital's rule to show $\lim\limits_{x\rightarrow0}\frac{\sin(x)}{x} = 1$ is circular reasoning, then how do we justify it still applies?
126701
How to prove that $d \sin(x)/dx = \cos(x)$ without circular logic such as L'Hôpital's rule?
150981
Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?
120493
L'hopitals rule with limits
26408
Evaluate a limit (probably involving L'Hôpital rule)
85249
Evaluating: $\lim\limits_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$
86984
Is L'Hopitals rule applicable to complex functions?
122301
Evaluating and proving $\lim\limits_{x\to\infty}\frac{\sin x}x$
138583
What is the limit of $\lim\limits_{x→∞}\frac{\sin x}{x}$
113781
How can I calculate $\lim_{x\to 0} \frac{\arctan(x)-\arcsin(x)}{x^3}$ without L'Hôpital?
122990
How can I prove that $\lim\limits_{n\to\infty}\frac{n+\sin n}{n+1} = 1$
68835
Trigonometric limit: Dividing the Numerator and Denominator by $\theta\;?$
30181
Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$
103055
Proving the quotient rule for limits
27423
Why does $\lim_{x\to 0^+} x^{\sin x}=1$?
106142
Why does L'Hopital's rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?
34127
Prove $\lim_{x \rightarrow 0} \frac {\sin(x)}{x} = 1$ with the epsilon-delta definition of limit.
106828
If $0 \lt x \lt \frac{\pi}{2}$, is $\cos(x) \le \frac{\sin(x)}{x} \le \frac{1}{\cos(x)}$?