Dowqueuestbew1j

2022-01-16

Questions about proving $\underset{x\to 0}{lim}\frac{\mathrm{sin}x}{x}=1$

Debbie Moore

Beginner2022-01-17Added 43 answers

A general result: let $f\left(x\right)=\sum _{n=0}^{\mathrm{\infty}}{a}_{n}{x}^{n}$ be a power series with radius of convergence R>0, where $R=\mathrm{\infty}$ is allowed.

If$0<r<R$ , then the power series converges uniformly on [−r,r]. Since the functions $a}_{n}{x}^{n$ are continuous, f is continuous on [−r,r]. Since r with 0<r<R was arbitrary, f is continuos on (−R,R).

In your example the sequence$\left({f}_{n}\right)$ does not converge uniformly.

If

In your example the sequence

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A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)Whether f is a function from Z to R if

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a) $f\left(n\right)=\pm n$.

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