x2-10x+21 Let f(x)= Complete parts (a) through (c) below. -3x a. Analyze lim f(x), lim f(x), lim f(x), and lilm f(x). x0- x-3 X-3* lim f(x)= (Simplify your answer.) lim f(x)= (Simpilify your answer.) X-0+ lim f(x)= (Simplify your answer.) X-3- lim f(x)= (Simplify your answer.) x-3* b. Does the graph of f have any vertical asymptotes? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A The graph of f has one vertical asymptote at (Type an equation.) OB. The graph of f has two vertical asymptotes. The leftmost asymptote is and the rightmost asymptote is (Type equations.) O C. The graph of f has no vertical asymptotes.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![# Exploring Limits and Vertical Asymptotes
Let \( f(x) = \frac{x^2 - 10x + 21}{x^2 - 3x} \). Complete parts (a) through (c) below:
### a. Analyze
\[ \lim_{{x \to 0^-}} f(x) \]
\[ \lim_{{x \to 0^+}} f(x) \]
\[ \lim_{{x \to -3}} f(x) \]
\[ \lim_{{x \to 3^+}} f(x) \]
1. \[ \lim_{{x \to 0^-}} f(x) = \boxed{} \](Simplify your answer.)
2. \[ \lim_{{x \to 0^+}} f(x) = \boxed{} \](Simplify your answer.)
3. \[ \lim_{{x \to -3}} f(x) = \boxed{} \](Simplify your answer.)
4. \[ \lim_{{x \to 3^+}} f(x) = \boxed{} \](Simplify your answer.)
### b. Does the graph of \( f \) have any vertical asymptotes? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The graph of \( f \) has one vertical asymptote at \( \boxed{} \). (Type an equation.)
B. The graph of \( f \) has two vertical asymptotes. The leftmost asymptote is \( \boxed{} \) and the rightmost asymptote is \( \boxed{} \). (Type equations.)
C. The graph of \( f \) has no vertical asymptotes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F573db276-7549-444e-9b95-52b12d1d154b%2F505444e1-476e-4c29-9dbb-b3c071ac4edd%2Fvyvdxx2_processed.jpeg&w=3840&q=75)
- Coordinates: \([-10, 10] \times [-10, 10]\)
- **B.**
- 
- Coordinates: \([-10, 10] \times [-10, 10]\)
- **C.**
- 
- Coordinates: \([-10, 10] \times [-10, 10]\)
- **D.**
- 
- Coordinates: \([-10, 10] \times [-10, 10]\)
### Error Analysis
Next, graph \( f \) and note any errors obtained with the graphing utility. Choose the correct graph from the options below:
#### Options:
- **A.**
- 
- This graph displays a partially correct representation but there might be some noted deviations.
- **B.**
- 
- Presents another variant of the function's graphing attempt.
- **C.**
- 
- Displays an alternative graph with specific deviations to evaluate.
- **D.**
- 
- This graph shows another representation with potential error considerations.
### Instructions:
1. Review each graph option carefully.
2. Select the graph that accurately represents the function \( f \) based on your graphing utility's output.
3. For the error analysis, choose the graph that most accurately depicts any errors present.
*Click to select your answer(s).*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F573db276-7549-444e-9b95-52b12d1d154b%2F505444e1-476e-4c29-9dbb-b3c071ac4edd%2Funzx2t_processed.jpeg&w=3840&q=75)
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