Explainer: Evaluating Limits Using Algebraic Techniques

In this explainer, we will learn how to use algebraic techniques like factorization to evaluate limits.

Specifically, we will cover the case where in seeking the limit limο—β†’οŒΊπ‘“(π‘₯),

  • the function is a rational function: 𝑓(π‘₯)=𝑃(π‘₯)𝑄(π‘₯), with 𝑃(π‘₯) and 𝑄(π‘₯) polynomials in π‘₯,
  • substitution fails because the quotient 𝑃(π‘Ž)𝑄(π‘Ž) gives the β€œindeterminate form” 00.

An example is lim→οŠͺπ‘₯βˆ’2π‘₯βˆ’56π‘₯βˆ’3π‘₯βˆ’4, where 𝑃(π‘₯)=π‘₯βˆ’2π‘₯βˆ’56 and 𝑄(π‘₯)=π‘₯βˆ’3π‘₯βˆ’4 give 𝑃(4)=64βˆ’(2)(4)βˆ’56=0 and 𝑄(4)=16βˆ’(3)(4)βˆ’4=0.

Since both 𝑃 and 𝑄 are polynomials, 𝑃(4)=𝑄(4)=0 means that (π‘₯βˆ’4) is a factor of each polynomial. So we can rewrite and divide these factors out: 𝑃(π‘₯)𝑄(π‘₯)=π‘₯βˆ’2π‘₯βˆ’56π‘₯βˆ’3π‘₯βˆ’4=(π‘₯βˆ’4)(π‘₯+4π‘₯+14)(π‘₯βˆ’4)(π‘₯+1)=π‘₯+4π‘₯+14π‘₯+1,π‘₯β‰ 1.providedthat

Now with the new function 𝑔(π‘₯)=π‘₯+4π‘₯+14π‘₯+1, we see that

  1. 𝑓(π‘₯)=𝑔(π‘₯) on the set of all numbers where both functions are defined, which is the set β„βˆ’{βˆ’1,4},
  2. the function 𝑔 is defined at π‘₯=4 (unlike 𝑓).

From (2), we can calculate the limit of 𝑔 at π‘₯=4 by substitution: lim→οŠͺοŠ¨π‘”(π‘₯)=𝑔(4)=(4)+4(4)+144+1=465.

Next, we use the following general principle.

Limits and Almost Equal Functions

Suppose that

  • functions 𝑓 and 𝑔 are equal at all points of an interval except π‘₯=π‘Ž,
  • 𝑔 has the limit 𝐿 as π‘₯ tends to π‘Ž.

Then, limο—β†’οŒΊπ‘“(π‘₯)=𝐿 also.

Now, fact (1) allows us to conclude that limlim→οŠͺ→οŠͺοŠ©οŠ¨π‘“(π‘₯)=π‘₯βˆ’2π‘₯βˆ’56π‘₯βˆ’3π‘₯βˆ’4=465.

Example 1: Finding the Limit of Rational Functions by Eliminating Common Factors

Find limο—β†’οŠ¨οŠ¨οŠ¨π‘₯βˆ’2π‘₯2π‘₯βˆ’6π‘₯+4.

Answer

Since (2)βˆ’2(2)=0 and 2(2)βˆ’6(2)+4=0 also, we need to factor out (π‘₯βˆ’2)π‘₯βˆ’2π‘₯2π‘₯βˆ’6π‘₯+4=(π‘₯βˆ’2)π‘₯(π‘₯βˆ’2)(2π‘₯βˆ’2)=π‘₯2π‘₯βˆ’2.

So limlimwhichcanbefoundbysubstitutionο—β†’οŠ¨οŠ¨οŠ¨ο—β†’οŠ¨π‘₯βˆ’2π‘₯2π‘₯βˆ’6π‘₯+4=π‘₯2π‘₯βˆ’2=22(2)βˆ’2=22=1.

The following factorization extends the difference of squares ο€Ήπ‘₯βˆ’π‘Žο…=(π‘₯βˆ’π‘Ž)(π‘₯+π‘Ž): π‘₯βˆ’π‘Ž=(π‘₯βˆ’π‘Ž)ο€Ήπ‘₯+π‘Žπ‘₯+π‘Žο…οŠ©οŠ©οŠ¨οŠ¨ and then ο€Ήπ‘₯βˆ’π‘Žο…=(π‘₯βˆ’π‘Ž)ο€Ήπ‘₯+π‘Žπ‘₯+π‘Žπ‘₯+π‘Žο…οŠͺοŠͺ and, more generally, π‘₯βˆ’π‘Ž=(π‘₯βˆ’π‘Ž)ο€Ήπ‘₯+π‘Žπ‘₯+π‘Žπ‘₯+β‹―+π‘Žπ‘₯+π‘Žο…οŠοŠοŠοŠ±οŠ§οŠοŠ±οŠ¨οŠ¨οŠοŠ±οŠ©οŠοŠ±οŠ¨οŠοŠ±οŠ§ which can be written as π‘₯βˆ’π‘Žπ‘₯βˆ’π‘Ž=π‘₯+π‘Žπ‘₯+π‘Žπ‘₯+β‹―+π‘Žπ‘₯+π‘Ž, from which we deduce that limο—β†’οŒΊοŠοŠοŠοŠ±οŠ§π‘₯βˆ’π‘Žπ‘₯βˆ’π‘Ž=π‘›π‘Ž since the right-hand side has 𝑛 terms.

This shortens potentially lengthy factorizations and has the following useful corollary: limο—β†’οŒΊοŠοŠο‰ο‰οŠοŠ±ο‰π‘₯βˆ’π‘Žπ‘₯βˆ’π‘Ž=π‘›π‘šπ‘Ž.

Example 2: Finding the Limit of Rational Functions by Eliminating Common Factors

Find limο—β†’οŠ¨οŠ©οŠ¨8π‘₯βˆ’64π‘₯βˆ’4.

Answer

Substitution gives 00 and we note the common factor (π‘₯βˆ’2). Factoring the 8 in the numerator, 8π‘₯βˆ’64π‘₯βˆ’4=8ο€Ήπ‘₯βˆ’8π‘₯βˆ’4, reveals powers of 2: 8=2 and 4=2 so this is =8ο€Ήπ‘₯βˆ’2π‘₯βˆ’2.

Now limlimlimο—β†’οŠ¨οŠ©οŠ¨ο—β†’οŠ¨οŠ©οŠ©οŠ¨οŠ¨ο—β†’οŠ¨οŠ©οŠ©οŠ¨οŠ¨οŠ©οŠ±οŠ¨8π‘₯βˆ’64π‘₯βˆ’4=8ο€Ύπ‘₯βˆ’2π‘₯βˆ’2=8π‘₯βˆ’2π‘₯βˆ’2=8ο€Ό322=24, using the fact that limο—β†’οŒΊοŠοŠο‰ο‰οŠοŠ±ο‰π‘₯βˆ’π‘Žπ‘₯βˆ’π‘Ž=π‘›π‘šπ‘Ž.

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