43. Where is [tex]f(r)[/tex] not continuous?
A. Nowhere
B. [tex]r = \sqrt{3}[/tex]
44. Find the horizontal asymptote of [tex]y = \frac{x+8 x^3}{x^2-2}[/tex]
A. Does not exist
B. y = 5
C. 4
D. -2
45. Find the horizontal asymptote of the rational function [tex]\frac{x^2+2 x+1}{x^2-x^2+4}[/tex]
A. 1
B. [tex]\frac{1}{8}[/tex]
C. 0
46. Horizontal asymptotes divide _____ into regions.
A. x-axis
B. y-axis
C. x intercept
D. y-intercept.
47. Determine where [tex]\frac{t^{-2}}{(t-3)(t+3)}[/tex] is not continuous.
A. {2,0}
B. {3,-3}
C. {-5,3}
D. {0,2}.
48. Evaluate [tex]\lim _{x \rightarrow \infty}\left(1-\frac{3}{x}\right)^x[/tex].
A. 3
B. [tex]e^3[/tex]
C. [tex]e^{-3}[/tex]
D. -3.
49. If [tex]x=t^2+1[/tex] and [tex]y=t^3+2[/tex], find [tex]\frac{d y}{d x}[/tex].
A. 2t
B. 3t
C. [tex]\frac{3}{2}[/tex]
D. [tex]\frac{3t}{2}[/tex]
Answer (1)
Here's a summary of the solutions:
Q43: f ( r ) = 3 r is continuous everywhere. Nowhere
Q44: y = x 2 − 1 x + 8 x 3 has no horizontal asymptote. Does not exist
Q45: 4 x 2 + 2 x + 1 has no horizontal asymptote. it.
Q46: Horizontal asymptotes divide the y -axis into regions.
Q47: − 8 t 2 1 is not continuous at 0
Q48: lim x → ∞ ( 1 − x 3 ) x = e − 3
Q49: Given x = t 2 + 1 and y = t 3 + 2 , d x d y = 2 3 t
Explanation
Problem Overview The question presents a series of independent math problems, numbered 43 through 49. Each problem will be addressed individually, providing a step-by-step solution for each.
Question 43 Solution Question 43: Determine where the function f ( r ) = 3 r is not continuous. Since f ( r ) = 3 r is a linear function, it is continuous everywhere. Therefore, the answer is Nowhere.
Question 44 Solution Question 44: Find the horizontal asymptote of y = x 2 − 2 2 x + 8 x 3 = x 2 − 1 x + 8 x 3 .
To find the horizontal asymptote, we examine the limit as x approaches infinity. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote.
Question 45 Solution Question 45: Find the horizontal asymptote of the rational function x 2 − x 2 + 4 x 2 + 2 x + 1 = 4 x 2 + 2 x + 1 .
To find the horizontal asymptote, we examine the limit as x approaches infinity. Since the function is 4 x 2 + 2 x + 1 = 4 1 x 2 + 2 1 x + 4 1 , there is no horizontal asymptote.
Question 46 Solution Question 46: Horizontal asymptotes divide into regions. Horizontal asymptotes are horizontal lines, which divide the y -axis into regions.
Question 47 Solution Question 47: Determine where the function ( 1 − 3 ) ( 1 + 3 ) t − 2 = t 2 ( − 2 ) ( 4 ) 1 = − 8 t 2 1 is not continuous. The function − 8 t 2 1 is not continuous when the denominator is zero, i.e., when t = 0 .
Question 48 Solution Question 48: Evaluate lim x → ∞ ( 1 − x 3 ) x .
We can use the fact that lim x → ∞ ( 1 + x a ) x = e a . Therefore, lim x → ∞ ( 1 − x 3 ) x = e − 3 .
Question 49 Solution Question 49: Given x = t 2 + 1 and y = t 3 + 2 , find d x d y .
We can find d t d x = 2 t and d t d y = 3 t 2 . Then, d x d y = d x / d t d y / d t = 2 t 3 t 2 = 2 3 t .
Final Answers The answers to the questions are:
A. Nowhere
A. Does not exist
C. it.
B. y -axis
D. {0,2}
C. e − 3
D. $\frac{3}{2}t
Examples
These problems cover fundamental concepts in calculus and precalculus, such as continuity, asymptotes, limits, and derivatives. These concepts are essential in various fields, including physics, engineering, and economics. For instance, understanding continuity is crucial in modeling physical phenomena, while asymptotes help analyze the behavior of functions at extreme values. Limits are the foundation of calculus, and derivatives are used to optimize processes and model rates of change.
