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., $x = 0$) to understand limits.

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

    • As $x$ approaches $0$ from the left, limit is $- ext{infinity}$; from the right, it's $+ ext{infinity}$.

  • 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 $x$ approaches infinity (or negative infinity).

Definitions of Asymptotes

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

  • Horizontal Asymptote: As $x$ 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 $f(x) = rac{5x - 7}{4x + 3}$, as $x o ext{infinity}$, divide numerator and denominator by $x$.

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

Trigonometric Functions and Asymptotes

  • Trigonometric functions (e.g., $tan(x)$) 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.