# Limit

### Limits

• $$\epsilon$$ method
• sequence $$S_n$$ goes to infinite $$L$$ is limit of $$S_n$$, iif there is $$N$$ satisfing $$S_n$$ $$-$$ $$L < \epsilon$$ in every $$n>N$$.
• Upper limit
• $$\lim_{n\to\infty}$$ $$sup(a_n)$$
• Lower limit
• $$-\lim_{n\to\infty}$$ $$inf(a_n)$$

### Limit of function

$$f_n$$ $$(x)$$ $$=x^n$$ ($$n \in N$$)
$$f_n$$ $$:(0,1) \rightarrow (0,1)$$

• Pointwise limit

$$\lim_{n\to\infty}$$ $$f_n$$ $$(x)$$ $$=f(x) = 0$$ for every $$x$$

for $$x=0.5$$,
sequence of $$f_n$$$$(0.5)$$ is $$\{0.5, 0.25, 0.125 ...\}$$
$$\lim_{n\to\infty}$$ $$f_n$$ $$(0.5)=0$$

• Uniform limit

$$\sup_{n}$$ $$\{\lvert f(x) - f_n(x) \rvert\ :x \in (0,1) \}$$
$$=$$ $$\sup_{n}$$ $$\{\lvert f_n(x) \rvert\ :x \in (0,1) \}$$

$$\sup_{1}$$ $$\{\lvert x \rvert :x \in (0,1) \}$$ $$=1$$
$$\sup_{2}$$ $$\{\lvert x^2 \rvert :x \in (0,1) \}$$ $$=1$$
$$\sup_{n\to\infty}$$ $$\{\lvert x^n \rvert :x \in (0,1) \}$$ $$=1$$

$$n>N$$ such as $$\epsilon$$ argument can not be determined because $$N$$ changes according to $$x$$ value.