5-28 Find the limit. limx →∞(e^-x+2 cos3 x)

Name: Problem 22, Chapter 2: Biocalculus Calculus for the Life Sciences
Uploaded: 2019-07-31T21:34:23-08:00
Duration: 3 min 38 s
Channel: Keith Cannon
Description: 5-28 Find the limit. limx →∞(e^-x+2 cos3 x)

Key Concepts

Exponential Decay

Exponential decay refers to a process or function that decreases at a rate proportional to its current value, often approaching zero as the variable increases. In limits, terms exhibiting exponential decay become negligible at infinity.

Oscillatory Functions

Oscillatory functions, such as the cosine function, do not settle to a single value as the variable increases but instead continue to oscillate between fixed bounds. This behavior can prevent the existence of a limit when the oscillation is not dampened.

Limits at Infinity

This concept involves evaluating the behavior of functions as the independent variable approaches infinity. It is a fundamental idea in calculus used to determine the end behavior of functions and assess whether they converge to a finite value or diverge.

Transcript

00:01 So for this, in this section, we're looking at the long -term behavior of functions.

00:06 So f of x as x goes to infinity.

00:08 In this case, our f of x is e to the minus x plus two cosine three x.

00:13 Now you might guess right up the bat that, hey, this is a sum.

00:16 So let's try to bring the limit in and take this limit of a sum and make it a sum of limits.

00:22 And that's a very good guess.

00:24 That's what we're going to do here.

00:25 So like i said, make it a sum of limits.

00:27 So x goes to infinity of e to minus x.

00:33 Plus the limit as x -cose -infinity of 2 cosine 3x.

00:50 And so let's recall that, as you might guess, this will converge because it will converge because remember that e to the x converges to infinity as x goes to infinity, and e to the minus x is 1 over e to the x.

01:10 So that looks like, this term up here looks like 1 over infinity which converges to zero.

01:19 So we knock out this first term here, and now we have to rely on the second one for something, if we want this to converge.

01:32 So before we even get technical with it, let's just think about what some constant times cosine 3x might look like, okay? what might this look like? because what does cosine look like? well, cosine, remember, it's a wave function that looks like this, it oscillates, up and down, up and down.

01:52 Really, that's not a very poor condition.

01:54 It oscillates down here, up here, down here, and it starts off...

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