L’hospital’s rule is used when an initial evaluation of a limit results in an indeterminate form such as<\/p>\n
##0\/0## or ##(+-oo)\/(+-oo)##<\/p>\n
If, after application of L’hospital’s rule, your limit evaluation produces another indeterminate form you apply L’ospital’s rule again.<\/p>\n
L’hospital’s rule states the if<\/p>\n
##lim_(x->c)(f(x))\/(g(x))=0\/0 or (+-oo)\/(+-oo)##<\/p>\n
Then you can find the limit by evaluating<\/p>\n
##lim_(x->c)(f'(x))\/(g'(x))##<\/p>\n
as long as ##f'(x)## and ##g'(x)## both exist and ##g'(x)!=0##<\/p>\n
Let’s look at an example.<\/p>\n
Suppose we want to evalauate<\/p>\n
##lim_(x->0)(cos x – 1)\/(x^2)##<\/p>\n
By direct substitution we get<\/p>\n
##lim_(x->0)(cos x – 1)\/(x^2)=(cos 0 – 1)\/(0^2)=(1-1)\/0=0\/0##<\/p>\n
Which is an indeterminate form so we apply L’hospital’s rule and get<\/p>\n
##lim_(x->0)(-sin x)\/(2x)=(-sin 0)\/(2*0)=0\/0##<\/p>\n
And we get the indeterminate form ##0\/0## again.<\/p>\n
Let’s apply L’hospital’s rule one more time (i.e. find the derivative of the new numerator and denominator and evaluate the limit again).<\/p>\n
##lim_(x->0)(-sin x)\/(2x)=lim_(x->0)(-cos x)\/2=(-1\/2)##<\/p>\n
As you can see in the image below (zoomed in to the problem point), there is a removable discontinuity (a “hole”) at x =0.<\/p>\n","protected":false},"excerpt":{"rendered":"
L’hospital’s rule is used when an initial evaluation of a limit results in an indeterminate form such as ##0\/0## or ##(+-oo)\/(+-oo)## If, after application of L’hospital’s rule, your limit evaluation produces another indeterminate form you apply L’ospital’s rule again. L’hospital’s Read More …<\/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-28195","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/28195","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=28195"}],"version-history":[{"count":0,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/28195\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/media?parent=28195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/categories?post=28195"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/tags?post=28195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}