Limits and Asymptotes Overview

Overview

This lecture discusses infinite limits, one-sided limits, vertical asymptotes, and how to determine the behavior of functions near undefined points.

The limit of ( 1/x ) as ( x \to 0^- ) is ( -\infty ); as ( x \to 0^+ ) is ( +\infty ); limit as ( x \to 0 ) does not exist.
For ( 1/x^2 ), both left and right limits as ( x \to 0 ) are ( +\infty ).
When a function's denominator approaches zero, check left and right limits to determine if the limit is ( +\infty ), ( -\infty ), or does not exist.
Example: ( \lim_{x \to 2^-} 1/(x-2) = -\infty ); ( \lim_{x \to 3^+} 5/(x-3) = +\infty ).
If limits from left and right differ, the two-sided limit does not exist.

Use direct substitution in the function if the denominator does not go to zero.
Factor when possible to simplify before substitution.
Example: ( \lim_{x \to -3} \frac{x+3}{x^2+x-6} ) simplifies to ( 1/5 ) after factoring and canceling.

( \lim_{x\to0} \tan x = 0 ) by direct substitution.
( \lim_{x\to\pi/2^-} \tan x = +\infty ); ( \lim_{x\to\pi/2^+} \tan x = -\infty ); two-sided limit does not exist.
( \lim_{x \to 0^+} \ln x = -\infty ); ( \lim_{x \to 0^-} \ln x ) does not exist.
( \lim_{x\to0^+} \csc x = +\infty ).

Set the denominator equal to zero to find the vertical asymptotes.
Factor denominators when necessary (example: ( x^2 - 4 = (x+2)(x-2) ); vertical asymptotes at ( x=2, x=-2 )).
If a factor cancels with the numerator, it creates a hole, not an asymptote.
Denominator factors with imaginary roots do not make real asymptotes or holes.

Limit — The value a function approaches as the input approaches a specific point.
Infinity (( \infty )) — Signifies increasing or decreasing without bound.
Vertical Asymptote — A line ( x = a ) where the function grows without bound; found by making the denominator zero.
Hole — A removable discontinuity where the factor cancels between numerator and denominator.

Practice finding limits and vertical asymptotes using substitution and factoring techniques.
Attempt additional problems involving one-sided limits and vertical asymptotes.
Review the behavior of logarithmic and trigonometric functions near undefined points.