MAT137Y5 Chapter Notes - Chapter squeeze theorem: Squeeze Theorem

• School: UTM
• Department: Mathematics
• Course: MAT137Y5
• Professor: Ali Mousavidehshikh

Document Summary

Everything you need to know about the squeeze theorems: This theorem is used to simplify limits that seem at first gla(cid:374)(cid:272)e to (cid:374)ot ha(cid:448)e a(cid:374) a(cid:374)s(cid:449)e(cid:396). It is used (cid:396)a(cid:396)ely i(cid:374) (cid:373)athe(cid:373)ati(cid:272)s, (cid:271)ut it"s i(cid:374)t(cid:396)odu(cid:272)ed i(cid:374) this (cid:272)ou(cid:396)se to give a better understanding of how to use the delta-epsilon paradigm for proving. First, the formal definition: (cid:1858),(cid:1859), :(cid:4666)(cid:1853),(cid:1854)(cid:4667) ,(cid:1855) (cid:4666)(cid:1853),(cid:1854)(cid:4667), (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1859)(cid:4666)(cid:1876)(cid:4667) (cid:4666)(cid:1876)(cid:4667) im (cid:1858)(cid:4666)(cid:1876)(cid:4667)== im (cid:4666)(cid:1876)(cid:4667) (cid:1858): (cid:1872) (cid:1857)(cid:1866): im (cid:1859)(cid:4666)(cid:1876)(cid:4667)= After looking at the limit paper, it should be quite clear what this signifies. Since f(x) and g(x) must be less and greater then h(x), respectfully, then when f(x) and h(x) are equal to l, the(cid:374) it"s clear that g(x) will also equal l. The proof of this relies on the definition: (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1859)(cid:4666)(cid:1876)(cid:4667) (cid:4666)(cid:1876)(cid:4667). (cid:1864)(cid:1857)(cid:1872) >(cid:882) (cid:1854)(cid:1857) (cid:1859)(cid:1874)(cid:1857)(cid:1866) (cid:1853)(cid:1866)(cid:1856) >(cid:882),(cid:1871). (cid:1872). (cid:882)< |(cid:1876) (cid:1855)|< |(cid:1858)(cid:4666)(cid:1876)(cid:4667) |< (cid:882)< |(cid:1876) (cid:1855)|< | (cid:4666)(cid:1876)(cid:4667) |< (cid:1853)(cid:1866)(cid:1856) In this step, we simply established the standard epsilon-delta definitions of the limit.