{"id":1092,"date":"2017-10-12T01:46:55","date_gmt":"2017-10-12T05:46:55","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1092"},"modified":"2017-10-15T01:12:49","modified_gmt":"2017-10-15T05:12:49","slug":"definitions-10102017","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/12\/definitions-10102017\/","title":{"rendered":"Definitions 10\/10\/2017"},"content":{"rendered":"

Definition of Domain : Given the function \"f we say that the domain of f is the set \"A\".<\/p>\n

As we discussed today in class the domain is not simply all the values that your function can take in but rather all the values that you decide your function will take in.<\/p>\n

Definition \"\infty is the limit- \"lim_{x means that for all \"M we can find a \"\delta such that whenever \"0<, it follows that \"f(x).<\/p>\n

An important part of the definition to note is the part that states for all \"M we can find a \"\delta. This line implies that there is a different value of \"\delta\" for each value of \"M\" which is why in class today we had to find \"\delta\" in terms of \"M\" in order to complete this proof.<\/p>\n

Definition 4.2.1 – Let\u00a0\"f, and let \"c\" be a limit point of the domain \"A\". We say that \"lim_{x provided that, for all \"\epsilon\", there exists a \"\delta such that whenever \"0< (and \"x) it follows that $\\mid f(x) -L \\mid < \\epsilon$.<\/p>\n

It is interesting to note the difference between the standard definition for functional limits and the definition for \"\infty\" as the limit. This difference is due to the fact that mathematical operations are still not well defined when using \"\infty\". The last part of the standard definition of functional limits uses the expression \"\mid. We would not want to use this expression where \"L if we are attempting to rigidly prove that a limit equals \"\infty\" when we cannot even properly define \"f(x).<\/p>\n

Theorem 4.2.3-\u00a0Given \"f: and limit point \"c, the following statements are equivalent.<\/p>\n

i) \"lim_{x<\/p>\n

ii) For all sequences \"(x_n) such that \"x_n, it follows that \"f(x_n).<\/p>\n

Since i and ii are equivalent this theorem is used to to define the divergence criterion for functional limits. Mainly the divergence criterion utilizes ii. It says that if two sequences \"x_n\" and \"y_n\"\u00a0exist for a function \"f\" such that\u00a0\"x_n (and c is a limit point),\u00a0\"y_n and\u00a0\"lim but if \"lim we can say\u00a0\"lim_{x does not exist. Here we see the usefulness of Theorem 4.2.3 because it allows us to split \"lim_{x into two requirements that we can then use to define divergence for functional limits.<\/p>\n","protected":false},"excerpt":{"rendered":"

Definition of Domain : Given the function we say that the domain of f is the set . As we discussed today in class the domain is not simply all the values that your function can take in but rather all the values that you decide your function will take in. Definition is the limit- […]<\/p>\n","protected":false},"author":3524,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58823],"tags":[],"class_list":["post-1092","post","type-post","status-publish","format-standard","hentry","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-hC","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1092","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3524"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1092"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1092\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1092"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1092"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1092"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}