{"id":308,"date":"2007-02-14T12:06:13","date_gmt":"2007-02-14T12:06:13","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/02\/14\/basics-limits\/"},"modified":"2007-02-14T12:06:13","modified_gmt":"2007-02-14T12:06:13","slug":"basics-limits","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/02\/14\/basics-limits\/","title":{"rendered":"Basics: Limits"},"content":{"rendered":"

One of the fundamental branches of modern math – differential and integral calculus – is based on the concept of limits<\/em>. In some ways, limits are a very intuitive concept – but the formalism of limits can be extremely confusing to many people.<\/p>\n

Limits are basically a tool that allows us to get a handle on certain kinds
\nof equations or series that involve some kind of infinity, or some kind of value that is almost<\/em> defined. The informal idea is very simple; the formalism is also<\/em> pretty simple, but it’s often obscured by so much jargon that it’s hard to relate it to the intuition.<\/p>\n

<\/p>\n

The use of limits for finding almost<\/em> defined values sounds tricky, but
\nit’s really pretty simple, and it makes for a very good illustration.<\/p>\n

Think of the simple function: f(x)=(x-1)\/(sqrt(x)-1). Just looking at it, you should be able to quickly see that its value at x=1 is undefined – f(1)=(1-1)\/(1-1)=0\/0.<\/p>\n

But let’s look at what happens as we get close to<\/em> x=1.<\/p>\n

f(2)=2.414. f(1.5)=2.22. f(1.2)=2.09. f(1.1)=2.05. f(1.01)=2.005… As we look at values of numbers greater than 1, but ever closer and closer to x=1, we can see that as x gets closer to 1, f(x) gets closer and closer to 2.<\/p>\n

The same thing happens from the other direction. f(0.5)=1.71. f(0.9)=1.95. f(0.99)=1.995….<\/p>\n

At exactly x=1, the value of the function is undefined – it’s a division by zero. But from either side of 1 – greater or lesser – the closer we get to x=1, the closer f(x) gets to 2. So we say that the limit<\/em> of f(x) as x approaches 1 = 2 – more traditionally written: “limx→1<\/sub>f(x)=2″.<\/p>\n

A simple example of managing infinity with limits is the equation f(x) = (1\/x)+4. As x gets larger, f(x) obviously gets closer and closer to 4. It never actually reaches<\/em> four – but it gets closer and closer. For any number ε, no matter how small, you can find some<\/em> value x so that f(x)<4+ε, and after that x, f(x) will always<\/em> be less than 4+ε. Epsilon can be 10-800<\/sup> – and there’s someplace where after some value x, f(x) is always less than 4+10-800<\/sup>.<\/p>\n

Which brings us at last to the formal definition of a limit. We’ll start with
\nthe case where we get infinitely close<\/em> to a real value. Given a real-valued function f(x), defined in an open interval around<\/em> a value p (but not necessarily at<\/em> P). Then limx→p<\/sub>f(x) (the limit of f(x) as x approaches p) = L if and only if for all ε>0, there exists some value δ>0 such that for all x where 0<|x-p|<δ, |f(x)-L|<ε.<\/p>\n

That’s really just restating what we did with the example. It’s just a formal way of saying that as x gets closer and closer to p, f(x) gets closer and closer to L. The ε and δ are names for a pair of decreasing values – as x gets closer to p, both ε and δ get closer and closer to 0. <\/p>\n

For dealing with infinity, as in our second example, the formal definition is:
\nGiven a real-valued function f, limx→∞<\/sub>f(x)=L if and only if for all ε>0, there exists some real number n such that for all x>n |f(x)-L|<ε.<\/p>\n

This is exactly the same kind of trick that we used in our example of a limit as x approaches infinity – no matter how small a value you pick for epsilon, there is some<\/em> point on the curve after which f(x) will never<\/em> be farther than ε away from L. <\/p>\n","protected":false},"excerpt":{"rendered":"

One of the fundamental branches of modern math – differential and integral calculus – is based on the concept of limits. In some ways, limits are a very intuitive concept – but the formalism of limits can be extremely confusing to many people. Limits are basically a tool that allows us to get a handle […]<\/p>\n","protected":false},"author":1,"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":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[74],"tags":[],"class_list":["post-308","post","type-post","status-publish","format-standard","hentry","category-basics"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-4Y","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/308","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=308"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/308\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=308"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=308"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=308"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}