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Find the limits of the following functions. of the following functions. Find an interval on which convergence is uniform and another on which it is not.

$f_n(x) = (x/2)^n +(1/x)^n$

$g_n(x)=nx/(2+5nx)$

For $h_n(x) = (n+x)/(4n+x)$ show it converges uniformly on $[0,N]$ for any $N \lt \infty$ but not uniformly on $[0,\infty)$.

My text only has the theorem regarding uniform continuity so any help as to what algorithm is used for these type of questions would be greatly helpful

Thank you

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Uniform continuity and uniform convergence are not the same thing. We say that a sequence of functions $f_n$ converge uniformly to a function $f$ on a set $E$ iff for all $\epsilon$ there exists an $n_0$ such that $$\sup_{x\in E}|f_n(x)-f(x)|<\epsilon$$ whenever $n\geq n_0$. Convergence then fails to be uniform if there is an $\epsilon$ such that for each $n$, there is an $x$ such that $|f_n(x)-f(x)|\geq\epsilon.$ Anyhow, you'll first want to figure out what the pointwise limit functions are--both how and where they are defined. Then you can start playing with uniform convergence. –  Cameron Buie Feb 13 '13 at 4:11

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Hint: Assuming your $f_n:\mathbb{R}\rightarrow\mathbb{R}$, you see your $f_n(x)$'s are not defined at $x=0$, now regarding limit function that does not also exist while $x\in(0,1)$ because $\lim_{n\rightarrow}x^n\rightarrow 0$ while $x\in (0,1)$, but at $x=1,x=2$ limit exist and it is equal to $1$, clearly for $x>2$ the limit also does not exist.Now Analyze yourself.

$g_n(x)$'s are not defined at point $x=\frac{-2}{5n},n\in\mathbb{N}$, at $x=0$ limit does not exist, at $x=-n$ limit exists and equal to $0$,Now Analyze yourself.

$h_n(x)$'s are not defined at points $x=-4n$, in other case divide denominator and numerator by $n$ and see that limit function is $\frac{1}{4}$ as $n\rightarrow\infty$

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