{"id":178038,"date":"2024-02-29T16:23:27","date_gmt":"2024-02-29T15:23:27","guid":{"rendered":"https:\/\/liora.io\/en\/?p=178038"},"modified":"2026-08-07T03:08:41","modified_gmt":"2026-08-07T02:08:41","slug":"limit-of-a-function-how-to-determine-it","status":"publish","type":"post","link":"https:\/\/liora.io\/en\/limit-of-a-function-how-to-determine-it","title":{"rendered":"Limit of a function: how to determine it?"},"content":{"rendered":"\n

All scientific disciplines are based on mathematics, and data science is no exception. When the problems to be solved are optimization problems, you need to know what the limit of a function is. In this article, you’ll discover how to determine the limit of a function.<\/strong><\/p>\n\n\n\n

Limit: Definition<\/h2>\n\n\n\n

The limit of a function f<\/em> is the value the function approaches when its argument approaches a certain value. Mathematically, we write limx<\/em>\u2192a<\/em><\/sub> f<\/em>(x<\/em>) = l<\/em><\/span>.<\/p>\n\n\n\n

We say that f<\/em> tends towards l<\/em> when x<\/em> tends towards a<\/em>. Depending on the case, a<\/em> and l<\/em> may be real numbers, or equal to \u00b1\u221e.<\/p>\n\n\n\n

Limits of standard functions<\/h2>\n\n\n\n

Real-valued finite limits<\/h3>\n\n\n\n

For example, for f<\/em>(x<\/em>) = x<\/em>2<\/sup><\/span>, the limit when x<\/em> tends towards 2 is 22<\/sup> = 4<\/span>.<\/p>\n\n\n\n

Infinite limits<\/h3>\n\n\n\n

When the function is well defined at a real number a<\/em> (it is said to be continuous at a<\/em>), then the limit at a<\/em> is exactly f<\/em>(a<\/em>)<\/span>.<\/p>\n\n\n\n

When the variable x<\/em> takes on very large values (positively or negatively), we say that x<\/em> tends towards more or less infinity. In this case, we distinguish between cases where f<\/em>(x<\/em>)<\/span> approaches a finite value and those where f<\/em>(x<\/em>)<\/span> moves away towards infinity. In the former case, f<\/em> is said to converge, and in the latter, f<\/em> to diverge.<\/p>\n\n\n\n

Below is a table summarizing the limits of common functions at infinity.<\/p>\n\n\n\n

\"Table<\/a><\/figure>\n\n\n\n

When a function admits a finite limit l<\/em> \u2208 \u211d<\/span> when x<\/em> \u2192 \u221e<\/span>, then we say that f<\/em> admits a horizontal asymptote of equation y<\/em> = l<\/em><\/span>. In other words, the curve moves closer and closer to the straight line.<\/p>\n\n\n\n

Infinite limits at a finite value<\/h3>\n\n\n\n
Function<\/th>Limit as x<\/em> \u2192 0<\/th><\/tr><\/thead>
f<\/em>(x<\/em>) = 1\/x<\/em><\/th>+\u221e if x<\/em> > 0; \u2212\u221e if x<\/em> < 0<\/td><\/tr>
f<\/em>(x<\/em>) = ln(x<\/em>)<\/th>\u2212\u221e<\/td><\/tr>
f<\/em>(x<\/em>) = \u221ax<\/em><\/th>0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

When limx<\/em>\u2192a<\/em><\/sub> f<\/em>(x<\/em>) = \u00b1\u221e<\/span>, with a<\/em> a real number, then the curve of f<\/em> is said to have a vertical asymptote at a<\/em>. In other words, the curve moves closer and closer to the straight line with equation x<\/em> = a<\/em><\/span>.<\/p>\n\n\n\n

Find out more about our Data Science training courses<\/a><\/div><\/div>\n\n\n\n

Limit operations<\/h2>\n\n\n\n

In most cases, it is possible to perform limit operations to determine the limit.<\/p>\n\n\n\n

Limit of a sum of two functions<\/p>\n\n\n\n

\"Formula<\/a><\/figure>\n\n\n\n

Limit of a product of two functions<\/p>\n\n\n\n

\"Formula<\/a><\/figure>\n\n\n\n

Limit of a quotient of two functions<\/p>\n\n\n\n

Limit<\/th>Case 1<\/th>Case 2<\/th>Case 3<\/th>Case 4<\/th>Case 5<\/th>Case 6<\/th>Case 7<\/th>Case 8<\/th>Case 9<\/th>Case 10<\/th>Case 11<\/th>Case 12<\/th>Case 13<\/th><\/tr><\/thead>
lim f<\/em>(x<\/em>)<\/th>L<\/em> \u2208 \u211d<\/td>L<\/em> \u2208 \u211d<\/td>L<\/em> > 0 or +\u221e<\/td>L<\/em> < 0 or \u2212\u221e<\/td>L<\/em> > 0 or +\u221e<\/td>L<\/em> < 0 or \u2212\u221e<\/td>+\u221e<\/td>+\u221e<\/td>0<\/td>+\u221e<\/td>+\u221e<\/td>0<\/td>+\u221e or \u2212\u221e<\/td><\/tr>
lim g<\/em>(x<\/em>)<\/th>L\u2032<\/em> \u2260 0<\/td>+\u221e or \u2212\u221e<\/td>0 with g<\/em>(x<\/em>) > 0<\/td>0 with g<\/em>(x<\/em>) > 0<\/td>0 with g<\/em>(x<\/em>) < 0<\/td>0 with g<\/em>(x<\/em>) < 0<\/td>L\u2032<\/em> > 0<\/td>L\u2032<\/em> < 0<\/td>+\u221e or \u2212\u221e<\/td>L\u2032<\/em> > 0<\/td>L\u2032<\/em> < 0<\/td>0<\/td>+\u221e or \u2212\u221e<\/td><\/tr>
lim f<\/em>(x<\/em>)\/g<\/em>(x<\/em>)<\/th>L<\/em>\/L\u2032<\/em><\/td>0<\/td>+\u221e<\/td>\u2212\u221e<\/td>\u2212\u221e<\/td>+\u221e<\/td>+\u221e<\/td>\u2212\u221e<\/td>Indeterminate form<\/td>+\u221e<\/td>\u2212\u221e<\/td>Indeterminate form<\/td>Indeterminate form<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Click on the tables to display them full-screen.<\/p>\n\n\n\n

Determining a limit in practice<\/h2>\n\n\n\n

To determine limits, it’s sometimes simpler to use other functions. Comparing functions allows you to compare their limits, when they exist. There are two main properties for comparing functions.<\/p>\n\n\n\n

Comparison theorem<\/h3>\n\n\n\n

Let f<\/em> and g<\/em> be two functions defined on an interval I<\/em> = [a<\/em>, +\u221e)<\/span>.<\/p>\n\n\n\n

Assume that for any x<\/em> \u2208 I<\/em>, f<\/em>(x<\/em>) \u2265 g<\/em>(x<\/em>)<\/span>. If limx<\/em>\u2192+\u221e<\/sub> g<\/em>(x<\/em>) = +\u221e<\/span>, then limx<\/em>\u2192+\u221e<\/sub> f<\/em>(x<\/em>) = +\u221e<\/span>. If limx<\/em>\u2192+\u221e<\/sub> f<\/em>(x<\/em>) = \u2212\u221e<\/span>, then limx<\/em>\u2192+\u221e<\/sub> g<\/em>(x<\/em>) = \u2212\u221e<\/span>.<\/p>\n\n\n\n

Gendarme theorem<\/h3>\n\n\n\n

Let f<\/em>, g<\/em> and h<\/em> be three functions defined on an interval I<\/em> = [a<\/em>, +\u221e)<\/span>.<\/p>\n\n\n\n

Assume that for any x<\/em> \u2208 I<\/em>, g<\/em>(x<\/em>) \u2264 f<\/em>(x<\/em>) \u2264 h<\/em>(x<\/em>)<\/span>. If g<\/em> and h<\/em> tend to the same limit l<\/em> in +\u221e, then f<\/em> also tends to l<\/em>.<\/p>\n\n\n\n

You now know what a limit is and how to calculate it. Calculating limits allows you, for example, to define the derivative of a function as the limit of a rate of increase. If you’d like to discover all the mathematical concepts involved in data science, we invite you to take a look at our curricula.<\/p>\n\n\n\n

Training in Data Science<\/a><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"

All scientific disciplines are based on mathematics, and data science is no exception. When the problems to be solved are optimization problems, you need to know what the limit of a function is. In this article, you’ll discover how to determine the limit of a function. Limit: Definition The limit of a function f is […]<\/p>\n","protected":false},"author":76,"featured_media":178043,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"_acf_changed":false,"editor_notices":[],"footnotes":""},"categories":[2433],"class_list":["post-178038","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-ai"],"acf":[],"_links":{"self":[{"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/posts\/178038","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/users\/76"}],"replies":[{"embeddable":true,"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/comments?post=178038"}],"version-history":[{"count":3,"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/posts\/178038\/revisions"}],"predecessor-version":[{"id":209669,"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/posts\/178038\/revisions\/209669"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/media\/178043"}],"wp:attachment":[{"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/media?parent=178038"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/liora.io\/en\/wp-json\/wp\/v2\/categories?post=178038"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}