Limits at Infinity & End-Behavior Summary

Reviewing Fraction Algebra & Common Denominator in Limit Problems

• Instructor begins with a worked limit that contains two rational expressions.

  • Example skeleton shown verbally: $\frac{5}{x} - \frac{5}{x-5}$ (exact numbers not written on board in excerpt).

• Key algebraic move: build a common denominator so the two fractions can be written as one.

  • Multiply each numerator by the factor missing from its denominator.

  • Resulting common denominator in the example: $x(x-5)$.

  • New combined numerator still carries the same “limit as $x\to a$” tag until evaluation.

• Emphasis on algorithmic practice:

  • “Practice and refine those skills.”

  • If unsure, re-work textbook examples that are written out step-by-step.

Proper Limit Notation

• Repeated reminder: keep writing the word “limit” until the value is actually substituted.

  • Prematurely dropping the limit symbol is a notational error.

  • Even after algebraic simplification, the expression is still a function, not yet a number.

Choosing Efficient Methods

• Mention of the “three-hour method” (nickname for full LCD approach) vs. a quicker partial-match approach.

  • If denominators already share a factor, it is faster to multiply only by the missing piece rather than build a full, cumbersome LCD.

  • Strategy becomes essential in timed quizzes/exams—pick the path that minimizes steps while staying clear to you.

Conceptual vs. Formal View of Limits

• The class so far has used an informal/conceptual approach.

• A formal (rigorous) ε–δ definition of limit is in the textbook and will be addressed at the end of the week.

• Goal: understand the concept first; formality comes later.

Introducing Limits at Infinity

• Until now, limits were finite: $2, 3, 0, -\tfrac12$, etc.

• Need to extend the idea to $x\to\infty$ and $x\to-\infty$.

• Notational note: because $\infty$ is “all the way to the right,” a one-sided symbol $\lim_{x\to\infty^-}$ makes no sense—you always approach $\infty$ from the left.

Informal Definition

• “As $x$ gets larger and larger (positively or negatively), what does $f(x)$ approach?”

• This is essentially end behavior analysis.

Example 1 – Linear End Behavior

• Evaluate $\lim_{x\to\infty} \bigl(-2x-3\bigr)$.

  • Table constructed with $x=10,100,1000,10000,10^6$.

  • Corresponding $y$ values: $-23,-203,-2003,-20003,-2\times10^{6}-3$.

  • Pattern: grows toward $-\infty$.

• Graphically, the line has negative slope, so the right end of the graph points down indefinitely.

• Concept of the dominating term: for large $|x|$, $-2x\gg -3$, so the constant is negligible.

• By symmetry, $\lim_{x\to-\infty}(-2x-3)=\infty$ (negative × negative → positive, then subtract 3).

General End Behavior of Polynomials

• Any polynomial $P(x)=a<em>nx^n+a</em>{n-1}x^{n-1}+\dots+a<em>0$ is controlled, for large $|x|$, by its leading term $a</em>nx^n$.

  • All smaller-degree terms become negligible.

• End behavior depends only on two features of that leading term:

  1. Degree parity (even vs. odd).

  2. Sign of the leading coefficient ($an>0$ or $an<0$).

Handy 2×2 Table (memorize or reconstruct)

• Visualization often drawn with small curves between the arrows; middle behavior is problem-specific.

Example 2 – High-Degree Polynomial

Evaluate $\lim_{x\to-\infty}\bigl(-3x^4-5x^2-9x+6\bigr).$

• Leading term: $-3x^4$ (even degree, negative sign) ⇒ both ends point down.

• Therefore limit is $-\infty$.

End Behavior of Rational Functions

Definition

• A rational function $R(x)=\dfrac{P(x)}{Q(x)}$ with

  • $m=\deg P(x),\; n=\deg Q(x).$

Degree-Comparison Rules (start of list; more cases promised next class)

1. Case $m

   • Example: $\dfrac{5}{x}$ with $m=0, n=1$.

   • As $x\to\pm\infty,$ the denominator grows faster; numerator is bounded.

   • Formal generalization: \lim_{x\to\infty}\frac{k}{x^{n}} = 0\quad (k\text{ constant},\;n>0).

   • Result: $\lim_{x\to\pm\infty}R(x)=0.$

2. Additional cases ($m=n$ and $m>n$) will be covered in the next lecture.

Conceptual Justification for Case $m<n$

• “The denominator dominates”; the fraction is squashed toward zero.

• Works even if denominator is a higher root power $x^{1/9}$ etc., as long as exponent is positive.

Practical Tips & Classroom Reminders

• Timed assessments: choose the fastest still-correct algebraic route; save “extra-step” methods for homework.

• Pattern recognition: more practice ⇒ quicker decisions on factoring vs. LCD vs. rationalizing vs. trig identities.

• Instructor humor: students asked for “thumbs-up,” “little finger at end of semester,” etc.—engagement keeps class lively.

Forthcoming Topics

• Completion of rational-function end-behavior chart (cases $m=n$ and $m>n$).

• Formal ε–δ definition of limits for those “sneaking ahead” in the book.

• Additional examples, including trigonometric and more complex rational limits.

Reviewing Fraction Algebra & Common Denominator in Limit Problems

• Instructor begins with a worked limit that contains two rational expressions.

• Example skeleton shown verbally: $\frac{5}{x} - \frac{5}{x-5}$ (exact numbers not written on board in excerpt).

• Key algebraic move: build a common denominator so the two fractions can be written as one.

• Multiply each numerator by the factor missing from its denominator.

• Resulting common denominator in the example: $x(x-5)$.

• New combined numerator still carries the same “limit as $x\to a$” tag until evaluation.

• Emphasis on algorithmic practice:

  • “Practice and refine those skills.”

  • If unsure, re-work textbook examples that are written out step-by-step.

Proper Limit Notation

• Repeated reminder: keep writing the word “limit” until the value is actually substituted.

• Prematurely dropping the limit symbol is a notational error.

• Even after algebraic simplification, the expression is still a function, not yet a number.

Choosing Efficient Methods

• Mention of the “three-hour method” (nickname for full LCD approach) vs. a quicker partial-match approach.

• If denominators already share a factor, it is faster to multiply only by the missing piece rather than build a full, cumbersome LCD.

• Strategy becomes essential in timed quizzes/exams—pick the path that minimizes steps while staying clear to you.

Conceptual vs. Formal View of Limits

• The class so far has used an informal/conceptual approach.

• A formal (rigorous) ε–δ definition of limit is in the textbook and will be addressed at the end of the week.

• Goal: understand the concept first; formality comes later.

Introducing Limits at Infinity

• Until now, limits were finite: $2, 3, 0, -\tfrac12$, etc.

• Need to extend the idea to $x\to\infty$ and $x\to-\infty$.

• Notational note: because $\infty$ is “all the way to the right,” a one-sided symbol $\lim_{x\to\infty^-}$ makes no sense—you always approach $\infty$ from the left.

Informal Definition

• “As $x$ gets larger and larger (positively or negatively), what does $f(x)$ approach?”

• This is essentially end behavior analysis.

Example 1 – Linear End Behavior

• Evaluate $\lim_{x\to\infty} \bigl(-2x-3\bigr)$.

• Table constructed with $x=10,100,1000,10000,10^6$ .

• Corresponding $y$ values: $-23,-203,-2003,-20003,-2\times10^{6}-3$.

• Pattern: grows toward $-\infty$.

• Graphically, the line has negative slope, so the right end of the graph points down indefinitely.

• Concept of the dominating term: for large $|x|$, $-2x\gg -3$ , so the constant is negligible.

• By symmetry, $\lim_{x\to-\infty}(-2x-3)=\infty$ (negative × negative → positive, then subtract 3).

General End Behavior of Polynomials

• Any polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ is controlled, for large $|x|$, by its leading term $a_nx^n$.

• All smaller-degree terms become negligible.

• End behavior depends only on two features of that leading term:

1. Degree parity (even vs. odd).

2. Sign of the leading coefficient ($a_n>0$ or $a_n<0$).

Handy 2×2 Table (memorize or reconstruct)

• Visualization often drawn with small curves between the arrows; middle behavior is problem-specific.

Example 2 – High-Degree Polynomial

Evaluate $\lim_{x\to-\infty}\bigl(-3x^4-5x^2-9x+6\bigr)$.

• Leading term: $-3x^4$ (even degree, negative sign) ⇒ both ends point down.

• Therefore limit is $-\infty$.

End Behavior of Rational Functions

Definition

• A rational function $R(x)=\dfrac{P(x)}{Q(x)}$ with

• $m=\deg P(x),\; n=\deg Q(x)$.

Degree-Comparison Rules (start of list; more cases promised next class)

1. Case $m

   • Example: $\dfrac{5}{x}$ with $m=0, n=1$.

   • As $x\to\pm\infty,$ the denominator grows faster; numerator is bounded.

   • Formal generalization: \lim_{x\to\infty}\frac{k}{x^{n}} = 0\quad (k\text{ constant},\;n>0).

   • Result: $\lim_{x\to\pm\infty}R(x)=0$.

   • Additional cases ($m=n$ and $m>n$) will be covered in the next lecture.

Conceptual Justification for Case $m<n$

• “The denominator dominates”; the fraction is squashed toward zero. Works even if a higher root power $x^{1/9}$ etc., as long exponent positive.

Practical Tips & Classroom Reminders

• Timed assessments: choose the fastest still-correct algebraic route; save “extra-step” methods for homework.

• Pattern recognition: more practice ⇒ quicker decisions on factoring vs. LCD vs. rationalizing vs. trig identities.

• Instructor humor: students asked for “thumbs-up,” “little finger at end of semester,” etc.—engagement keeps class lively.

Forthcoming Topics

• Completion of rational-function end-behavior chart (cases $m=n$ and $m>n$).

• Formal ε–δ definition of limits for those “sneaking ahead” in the book.

• Additional examples, including trigonometric and more complex rational limits.