Find the limit at a vertical asymptote of a rational function II

Learning Objectives Determine the one-sided limits (from the left and right) of a rational function at a vertical asymptote. Analyze the signs of the numerator and denominator as x approaches a vertical asymptote from each side. Conclude whether the limit at a vertical asymptote is positive infinity, negative infinity, or does not exist. Differentiate between vertical asymptotes and removable discontinuities by analyzing common factors. Evaluate limits at vertical asymptotes where the denominator has a factor raised to an even or odd power. Apply the concept of one-sided limits to sketch the behavior of a rational function's graph near its vertical asymptotes. What happens when you divide by a number that's almost zero? Does it matter if it's a tiny positive o...

TermDefinitionExample Vertical Asymptote (VA)A vertical line x = c where the function f(x) approaches positive or negative infinity as x approaches c from the left or right. It occurs where the denominator of a simplified rational function is zero.The function f(x) = 1/(x-2) has a vertical asymptote at x=2. One-Sided LimitThe value a function approaches as x approaches a point from only one side. It is denoted as lim x→c⁻ f(x) (from the left) and lim x→c⁺ f(x) (from the right).For f(x) = 1/x, the limit as x approaches 0 from the right is lim x→0⁺ (1/x) = ∞. Infinite LimitA limit in which the function's value increases or decreases without bound as x approaches a certain number. The result is written as ∞ or -∞.The limit of f(x) = 1/x² as x approaches 0 is ∞. Sign AnalysisThe process...

One-Sided Limit Sign Analysis To find \lim_{x \to c^+} f(x) or \lim_{x \to c^-} f(x): \newline 1. Determine the sign of the Numerator as x → c. \newline 2. Determine the sign of the Denominator as x approaches c from the specified side. \newline 3. Use sign rules: \frac{(+)}{(+)} \to +\infty, \frac{(+)}{(-)} \to -\infty, \frac{(-)}{(+)} \to -\infty, \frac{(-)}{(-)} \to +\infty. This is the primary method for determining the behavior at a vertical asymptote. Use a test value very close to 'c' on the appropriate side (e.g., c+0.1 for c⁺, c-0.1 for c⁻) to find the signs. Even/Odd Power Rule at Asymptotes Consider the factor (x-c)^n in the simplified denominator that causes the asymptote. \newline If n is even, \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x). (Behavior is...