Notes on Limits Involving Infinity and Asymptotes

Introduction

  • Lesson overview: limits involving infinity and asymptotes.

  • Learning objectives:

    • Find infinite limits and limits at infinity.

    • Determine horizontal, vertical, and slant asymptotes using limits.

  • Keywords introduced: finite limit, limit at infinity, vertical asymptote, horizontal asymptote, slant asymptote.

Basic Concepts of Limits

  • Limits can often be calculated using a shortcut method previously discussed in earlier lessons.

  • Example problem: Rational function behavior as it approaches a singularity.

  • Graphing a function can illustrate key points, specifically around undefined behaviors.

Infinite Limits

  • Analyzing behavior near points of discontinuity:

    • Substitute values close to the point of discontinuity (e.g., x=0x = 0) to understand limits.

    • Calculate at values like −0.01-0.01 and 0.010.01 to determine behavior at that point:

    • As xx approaches 00 from the left, limit is −extinfinity- ext{infinity}; from the right, it's +extinfinity+ ext{infinity}.

  • Vertical asymptote created where limit does not exist due to mismatched limits from either side.

Vertical and Horizontal Asymptotes

  • Vertical asymptotes occur when limits approach infinity from either side but are not equal.

  • Horizontal asymptotes depend on the growth rates of the numerator and denominator as xx approaches infinity (or negative infinity).

Definitions of Asymptotes

  • Vertical Asymptote: Graph approaches a line, may not intersect.

  • Horizontal Asymptote: As xx tends to $ ext{infinity}$, the output approaches a constant value.

  • Slant Asymptote: Occurs when the degree of the numerator is one higher than that of the denominator; found via polynomial long division.

Indeterminate Forms and Limits

  • Indeterminate form examples: $ rac{ ext{infinity}}{ ext{infinity}}$ implies deeper analysis needed:

    • Take the highest power in the denominator and divide through the entire equation.

  • Application related to indeterminate forms:

    • Example limit: from rational functions, apply division of the highest degree.

Calculating Specific Limits

  • Evaluation of limits can be done via substitution:

    • Example: For f(x)=rac5x−74x+3f(x) = rac{5x - 7}{4x + 3}, as xoextinfinityx o ext{infinity}, divide numerator and denominator by xx.

    • Limits around +extinfinity+ ext{infinity} result in constant values, indicating horizontal asymptotes.

Trigonometric Functions and Asymptotes

  • Trigonometric functions (e.g., tan(x)tan(x)) have vertical asymptotes at intervals of $ rac{ ext{pi}}{2} + n ext{pi}$.

  • Consider limits approaching these critical points to identify asymptotic behavior.

Real-World Applications

  • Example problem on the diameter of an animal's pupil based on light intensity. This connects earlier theoretical knowledge to practical application.

  • Minimum light results in pupil dilation (x approaching 0); maximum light causes constriction (x approaching infinity).

Conclusion

  • Importance of limits in understanding function behavior near particular values and inferring properties globally.

  • Preparing for further lessons by revisiting foundational lesson concepts.