5-28 Find the limit. limx →∞(√(x^2+a x)-√(x^2+b x)) |...

Name: Problem 18, Chapter 2: Biocalculus Calculus for the Life Sciences
Uploaded: 2019-07-31T21:13:39-08:00
Duration: 10 min 16 s
Channel: Keith Cannon
Description: 5-28 Find the limit. limx →∞(√(x^2+a x)-√(x^2+b x))

Key Concepts

Limits at Infinity

Limits at infinity involve determining the behavior of functions as the variable grows without bound. In these situations, it is important to identify which terms in an expression dominate the behavior of the function, allowing for simplifications that capture the asymptotic behavior of the function as the variable becomes very large.

Asymptotic Analysis

Asymptotic analysis is the process of approximating functions in limiting cases, such as for large values of a variable. By focusing on the leading terms of an expression, one can simplify complex expressions and identify the principal contributions that determine the limiting behavior, which is particularly useful when dealing with functions involving radicals or other non-linear operations.

Algebraic Manipulation of Radical Expressions

Algebraic manipulation of radical expressions often involves rewriting or simplifying expressions that contain square roots or other radicals. This can include factoring out dominant terms or rewriting the expression to reveal its leading order behavior, thereby making it easier to analyze or compute limits.

Rationalization Technique

The rationalization technique is used to eliminate radicals from the numerator or denominator of an expression, typically by multiplying by the conjugate. This method is particularly effective in dealing with differences of square roots, as it can transform the expression into a form where cancellation and limit evaluation become simpler.

Transcript

00:03 Okay, so with this problem, we're again asked to look at the long -term behavior of this function of x.

00:08 And like many others, it's not clear to tell what it will look like.

00:11 It's not x squared or x or something like that or a over x, so we can't really tell if it's going to converge immediately or diverge or to what.

00:19 And so our first task will be to simplify it a little bit and hopefully cancel some turns.

00:24 In other problems, we've done so by multiplying or dividing by some sort of quotient by the highest power in the denominator or other problems.

00:33 Other things like that, or by employing some limit arithmetic rules where a limit of a sum of sequences is equal to the sum of limits of sequences and others that you've seen in 2 .1.

00:45 So let's take this off in a gear immediately by employing a trick called multiplying by the radical conjugate, where we multiply this by a term that's equal to one, but we are flipping the sign in the middle here.

00:58 So if you had a term square root of a plus square root of b, it's radical.

01:03 Conjugate it would be square of a minus square root of b and so what we what we're multiplying by is square root of x squared the square roots of x squared plus a squared uh plus a x sorry plus not minus plus square roots of x squared plus b over square root x squared plus a x squared plus a x plus a x plus square root of x squared plus bx now the denominator won't change our our whole term will end up looking something like this our denominator won't change because this isn't really a question here it's just over one but we will end up simplifying this term quite a bit and our numerator will be something much simpler as we'll see because we are multiplying if we foil this out we get square of x squared plus a x times squared of x squared plus a x times squared of x squared plus a x which is just going to look like x squared plus a x.

02:21 And since we're foiling here, once we do outer inner, we'll see that we are multiplying square root of x squared plus a x times a square root of x squared plus bx minus square root of x squared plus bx times the square root of x squared plus a x and that's of the form of something like, you know, a .b.

02:49 Minus b .a.

02:49 So it just cancels.

02:51 Okay.

02:51 So then our last term will be minus x squared plus bx all over square roots of x squared, square root of x squared plus ax plus square root of x squared plus bx.

03:19 Okay.

03:21 And so keep simplifying the numerator a little bit, and we'll see that on top we'll get limit as x -cost infinity of a minus b times x.

03:38 The x squared's cancel, and then you just factor out the x, and then we have that all over this numerator x squared plus a x plus square of x squared plus bx...

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