{"id":55102,"date":"2018-02-20T23:50:30","date_gmt":"2018-02-20T23:50:30","guid":{"rendered":"https:\/\/writemyessayfree.com\/how-do-you-use-the-definition-of-continuity-and-the-properties-of-limits-to-show-that-the-function-gx-sqrt-x2-8x-15-is-continuous-on-the-interval-35-2"},"modified":"2017-08-16T13:19:13","modified_gmt":"2017-08-16T13:19:13","slug":"how-do-you-use-the-definition-of-continuity-and-the-properties-of-limits-to-show-that-the-function-gx-sqrt-x2-8x-15-is-continuous-on-the-interval-35-2","status":"publish","type":"post","link":"https:\/\/www.benedictsol.com\/blogs\/how-do-you-use-the-definition-of-continuity-and-the-properties-of-limits-to-show-that-the-function-gx-sqrt-x2-8x-15-is-continuous-on-the-interval-35-2\/","title":{"rendered":"How do you use the definition of continuity and the properties of limits to show that the function ##g(x) = sqrt(-x^2 + 8*x &#8211; 15)## is continuous on the interval [3,5]?"},"content":{"rendered":"<p>There is no one sentence answer.<\/p>\n<p>In order for ##g## to be continuous on ##[3,5]##, the definition of continuous on a closed interval requires:<\/p>\n<p>For ##c## in ##(3,5)##, we need##lim_(xrarrc) g(x) = g(c)## and we also need one-sided continuity at the endpoints: we need: ## lim_(xrarr3^+) g(x) = g(3)## and ##lim_(xrarr5^-) g(x) = g(5)##<\/p>\n<p>For ##c## in ##(3,5)##, We&#8217;ll use the properties of limits to evaluate the limit:<\/p>\n<p>##lim_(xrarrc) g(x) = lim_(xrarrc) sqrt(-x^2+8x-15)##<\/p>\n<p>##= sqrt(lim_(xrarrc)(-x^2+8x-15))##<\/p>\n<p>##= sqrt(lim_(xrarrc)(-x^2)+lim_(xrarrc)(8x)-lim_(xrarrc)(15))##<\/p>\n<p>##= sqrt(-lim_(xrarrc)(x^2)+8lim_(xrarrc)(x)-lim_(xrarrc)(15))##<\/p>\n<p>##= sqrt(-(lim_(xrarrc)(x))^2+8lim_(xrarrc)(x)-lim_(xrarrc)(15))##<\/p>\n<p>##= sqrt(-(c)^2+8(c)-(15))##<\/p>\n<p>##= g(c)##<\/p>\n<p>Use the one-sided versions of the limit properties at the endpoints.<\/p>\n<p>For ##c=3##, replace all limits of the form ##lim_(xrarrc)## with ##lim_(xrarr3^+)##<\/p>\n<p>For ##c=5##, replace all limits of the form ##lim_(xrarrc)## with ##lim_(xrarr5^-)##<\/p>\n","protected":false},"excerpt":{"rendered":"<p>There is no one sentence answer. In order for ##g## to be continuous on ##[3,5]##, the definition of continuous on a closed interval requires: For ##c## in ##(3,5)##, we need##lim_(xrarrc) g(x) = g(c)## and we also need one-sided continuity at <a href=\"https:\/\/www.benedictsol.com\/blogs\/how-do-you-use-the-definition-of-continuity-and-the-properties-of-limits-to-show-that-the-function-gx-sqrt-x2-8x-15-is-continuous-on-the-interval-35-2\/\" class=\"read-more\">Read More &#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-55102","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/55102","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/comments?post=55102"}],"version-history":[{"count":0,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/55102\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/media?parent=55102"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/categories?post=55102"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/tags?post=55102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}