Basic Mathematics · Unit 3
Asymptotes and Curve Behavior
Exam-focused notes for Asymptotes and Curve Behavior (Basic Mathematics, MTH104): what the TU syllabus asks and how it has actually been tested, with 3 solved past questions from this unit.
What this unit covers
- Horizontal asymptotes
- Vertical asymptotes
- Oblique asymptotes
- Asymptotic behavior analysis
Horizontal asymptotes
Define horizontal and vertical asymptotes. Find the appropriate asymptotes to the function: $f(x) = x - \sqrt{x^2 + 16}$. [2+3]
- Function: $f(x) = x - \sqrt{x^2 + 16}$ - Domain check: $x^2 + 16 0$ for all real $x$, so domain is all of $\mathbb{R}$. --- (a) Definitions Horizontal Asymptote: A line $y = L$ is a horizontal asymptote of $y = f(x)$ if $$\lim{x \to +\infty} f(x) = L \qua...
Full solved answer →Asymptotes: Definition and Types
- Function: $f(x) = \dfrac{-8}{x^2 - 4}$ - Numerator: constant $-8$ (degree 0) - Denominator: $x^2 - 4$ (degree 2) - Required: definition of asymptotes, types, horizontal and vertical asymptotes, existence of other asymptotes. All required data is present. ...
Full solved answer →Define horizontal asymptote to a curve $y = f(x)$. Find the horizontal asymptote to the curve $f(x) = \frac{5x^2 + 8x - 3}{3x^2 + 2}$ and draw the curve. [2+3]
A line $y = L$ is called a horizontal asymptote to the curve $y = f(x)$ if the function approaches $L$ as $x$ tends to infinity in magnitude, i.e. $$\lim{x \to +\infty} f(x) = L \qquad \text{or} \qquad \lim{x \to -\infty} f(x) = L.$$ As $x \to \infty$, the ...
Full solved answer →Make Unit 3 stick
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