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17calculus > limits > infinite limits

Infinite Limits - Limits At Infinity

Finite Limits, Infinite Limits, Limits At Infinity . . . Terminology Explained

The use of the terms finite limits, infinite limits and limits at infinity are used differently in various books and your instructor may have their own idea of what they mean. In this panel, we will try to break down the cases and explain the various ways these terms can be used as well as how we use them here at 17calculus.

When we talk about limits, we are looking at the \(\displaystyle{ \lim_{x \to c}{f(x)} = L }\). The various terms apply to the description of \(c\) and \(L\) and are shown in the table below. The confusion lies with the terms finite limits and infinite limits. They can mean two different things.

\(\displaystyle{ \lim_{x \to c}{f(x)} = L }\)

when

term(s) used

\(c\) is finite

limits approaching a finite value or finite limits

\(c\) is infinite \(\pm \infty\)

limits at infinity or infinite limits

\(L\) is finite

finite limits

\(L\) is infinite \(\pm \infty\)

infinite limits

You can see where the confusion lies. The terms finite limits and infinite limits are used to mean two different things, referring to either \(c\) or \(L\). It is possible to have \(c = \infty\) and \(L\) be finite. So is this an infinite limit or a finite limit? It depends if you are talking about \(c\) or \(L\).

How 17calculus Uses These Terms
The pages on this site are constructed based on what \(c\) is, i.e. we use the terms finite limits and infinite limits based on the value of \(c\) only ( using the first two rows of the table above and ignoring the last two ). This seems to be the best way since, when we are given a problem, we can't tell what \(L\) is until we finish the problem, and therefore we are unable to determine what type of problem we have and know what techniques to use until we are done with the problem.

Important: Make sure and check with your instructor to see how they use these terms.

This Page - Infinite Limits (or they may be called Limits At Infinity in your textbook) refers to cases where the variable in question goes off to infinity. In limit notation, they look like \(\displaystyle{ \lim_{x \rightarrow c}{~f(x)} }\) where c is \( \infty \) or \( -\infty \).

If your limit shows c as a finite number, then you need to go to the finite limits page. ( The panel above explains the terminology and how 17calculus defines finite and infinite limits. )

When evaluating \(\displaystyle{ \lim_{x \to \pm \infty}{~f(x)} }\) you need to determine if the graph of the function is leveling off at a value ( and, if so, what that value is ) or if it is going off to infinity ( either \(+\infty\) or \(-\infty\) ). You don't want to try to figure it out off a graph. You need to do it mathematically ( from the equation ).

This is the main theorem you will use.

Infinite Limits Theorem

\(\displaystyle{ \lim_{x \rightarrow \pm \infty}{\left[ \frac{1}{x} \right]} = 0 }\)

You can use the limit laws to apply this theorem to the case when you have \(\displaystyle{ \lim_{x \to \infty}{\left[ \frac{a}{x^k}\right]} }\) where \(k\) is a positive rational number and \(a\) is a real number. Here is an example. Try it on your own before looking at the solution.

Evaluate \(\displaystyle{ \lim_{x\to\infty}{\frac{3}{x^2}} }\).

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Practice Problems

Instructions - Unless otherwise instructed, evaluate the following limits, giving your answers in exact terms.

Level A - Basic

Practice A01

\(\displaystyle{\lim_{x\to\infty}{(x^4+7x^2+3)}}\)

answer

solution

Practice A02

\(\displaystyle{\lim_{x\to\infty}{(x^5-3x^2+x-21)}}\)

answer

solution

Practice A03

\(\displaystyle{\lim_{x\to\infty}{\frac{3x^2+5x+4}{x^3+7x}}}\)

solution

Practice A04

\(\displaystyle{\lim_{x\to-\infty}{\left[\frac{x+5}{3x+7}\right]}}\)

answer

solution

Practice A05

\(\displaystyle{\lim_{x\to-\infty}{\frac{7}{x^3-16}}}\)

answer

solution

Practice A06

\(\displaystyle{\lim_{x\to-\infty}{\frac{x^4+x}{5x^3+7}}}\)

solution

Practice A07

\(\displaystyle{\lim_{x\to-\infty}{\left[x-\sqrt{x^2+9}\right]}}\)

answer

solution

Practice A08

\(\displaystyle{\lim_{x\to\infty}{(3x^3-17x^2)}}\)

answer

solution

Practice A09

\(\displaystyle{\lim_{x\to\infty}{\frac{3}{x^2+5}}}\)

answer

solution

Practice A10

\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^3+x+12}{2x^3-5x}\right]}}\)

answer

solution

Practice A11

\(\displaystyle{ \lim_{x \to \infty}{\left[ \frac{7x^2 - 3x + 12}{x^3 + 4x + 127} \right]} }\)

answer

solution

Practice A12

\(\displaystyle{\lim_{x\to-\infty}{\left[\frac{7x^2+x+21}{11-x}\right]}}\)

answer

solution

Practice A13

\(\displaystyle{\lim_{x\to\infty}{\frac{4x^{10}+10000x^9}{5x^{10}+4}}}\)

solution

Practice A14

\(\displaystyle{\lim_{x\to\infty}{\frac{3x^7}{5x^8+10x+2}}}\)

solution

Practice A15

\(\displaystyle{\lim_{x\to\infty}{\frac{x^4}{x^3+5}}}\)

solution


Level B - Intermediate

Practice B01

\(\displaystyle{\lim_{x\to\infty}{\frac{x+3}{\sqrt{x^2+4}}}}\)

solution

Practice B02

\(\displaystyle{\lim_{x\to\infty}{\left[\sqrt{x^2+4x+1}-x\right]}}\)

solution

Practice B03

\(\displaystyle{ \lim_{x\to\infty}{\arctan(x)}}\)

answer

solution

Practice B04

\(\displaystyle{\lim_{x\to\infty}{\left[x-\sqrt{x^2+9}\right]}}\)

answer

solution

Practice B05

\(\displaystyle{\lim_{x\to\infty}{\sqrt{\frac{x^3+3x}{4x^3+7}}}}\)

answer

solution


Level C - Advanced

Practice C01

\(\displaystyle{\lim_{x\to-\infty}{\frac{x^3}{\sqrt{x^6+4}}}}\)

solution

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