If limx →0 (f(x))/(x^2)=1, find a. limx →0 f(x) b. limx →0 (f(x))/(x)

Name: Problem 80, Chapter 2: Thomas Calculus
Uploaded: 2021-01-23T02:05:43-08:00
Duration: 3 min 45 s
Channel: Madi Sousa
Description: If limx →0 (f(x))/(x^2)=1, find a. limx →0 f(x) b. limx →0 (f(x))/(x)

step 1 In this problem, we are given a limit that has a solution. We know the solution. And from there,

If limx →0 (f(x))/(x^2)=1, find a. limx →0 f(x) b. limx →0...

Key Concepts

Limit

A limit is a fundamental concept in calculus that describes the value a function approaches as its input approaches a particular point. Limits are used to analyze the behavior of functions near points of interest and form the basis for definitions of continuity, derivatives, and integrals.

Asymptotic Behavior

Asymptotic behavior concerns how a function behaves in the vicinity of a specific point, often comparing its growth rate to that of another function. When we say a function f(x) is asymptotically equivalent to x² (i.e., f(x)/x² ? 1 as x ? 0), we mean that near x = 0, f(x) behaves like x².

Order of Vanishing

The order of vanishing of a function at a point refers to the power of the variable that best describes how rapidly the function approaches zero. For instance, if f(x) behaves like x² as x tends to 0, it is said to vanish to second order, which is crucial when comparing the relative rates of decay in limit computations.

Transcript

00:01 In this problem, we are given a limit that has a solution.

00:05 We know the solution.

00:07 And from there, we need to figure out the solution to two other limits.

00:12 So really what we're dealing with here is beginning to understand how we can use limits across products and quotients of functions.

00:21 And we're going to be talking about that if you're cute.

00:24 Pardon me, if you're confused now, hopefully this will make sense at the end of the video.

00:29 So let's first start out with what we're given.

00:32 We're given the limit as x approach 0 of f of x over x squared, and we know this limit is 1.

00:38 So for part a is saying, well, let's try to find the limit as x approach 0 of f of x.

00:45 This is not the same as the limit before, but can i find this limit knowing that the first one equals 1? and the answer is yes, we can.

00:55 The first thing that we're going to do is we'll distribute this limit, in the numerator and the denominator of our given function.

01:03 So we'll get the limit as x approaches 0 of f of x divided by the limit as x approach 0 of x squared, and we know this equals 1.

01:14 So clearly we can see that if our x is approaching 0, our denominator is going to approach 0 as well.

01:20 What's 0 squared? well that's simply 0.

01:23 So if we rearrange this, we would say the limit as x approaches 0 of f of x equals 1 times 0.

01:30 We just moved this denominator to the other side.

01:35 So clearly we can see that the limit as x approach is 0 of f of x is 0.

01:41 And now on to part b, part b is a very similar procedure...

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