22. Find the range of [tex]f(x)=\frac{x-2}{x-3}[/tex].
A. [tex]R \{0,1\}[/tex]
B. [tex]R[/tex]
C. [tex]R \{2,3\}[/tex]
D. [tex]R \{1\}[/tex].
23. The [tex]y[/tex]-intercept of [tex]y=a x^3+b x^2+c x+d[/tex] is
A. [tex](y(0), 0)[/tex]
B. [tex](y(0), y(0))[/tex]
C. [tex](0,0)[/tex]
D. [tex](0, y(0))[/tex].
24. Evaluate [tex]\lim _{h \rightarrow 0} \frac{1}{h} \ln \left(\frac{2+h}{2}\right)[/tex].
A. [tex]e^2[/tex]
B. 1
C. [tex]\frac{1}{2}[/tex]
D. 0.
25. Evaluate the left-hand limit [tex]\lim _{x \rightarrow-3^{-}}\left(-\frac{5}{x+3}\right)[/tex].
A. [tex]-\infty[/tex]
B. [tex]x[/tex]
C. 2
D. -2 .
26. Evaluate [tex]\lim _{x \rightarrow-2} \frac{x+2}{\ln (x+3)}[/tex].
A. Does not exist
B. 2
C. 0
D. 1.
27. A function [tex]f(x)[/tex] is not continuous at [tex]a[/tex] if ...
A. [tex]f(a)=0[/tex]
B. [tex]f(a)[/tex] exists
C. [tex]f(a) \neq 0[/tex]
D. [tex]f(a)[/tex] is undefined.
Answer (1)
Find the range of f ( x ) = x − 3 x − 2 . The range is R \ { 1 } . D
The y -intercept of y = a x 3 + b x 2 + c x + d is ( 0 , y ( 0 )) . D
Evaluate lim h → 0 h 1 ln ( 2 2 + h ) . The limit is 2 1 . C
Evaluate the left-hand limit lim x → − 3 − ( − x + 3 5 ) . The limit is ∞ . A
Evaluate lim x → − 2 l n ( x + 3 ) x + 2 . The limit is 1. D
A function f ( x ) is not continuous at a if f ( a ) is undefined. D
Explanation
Introduction We will solve each multiple-choice question step-by-step, providing explanations and selecting the correct answer.
Question 22 Solution Question 22: Find the range of f ( x ) = x − 3 x − 2 .
To find the range, we set y = x − 3 x − 2 and solve for x in terms of y .
y ( x − 3 ) = x − 2 ⟹ y x − 3 y = x − 2 ⟹ y x − x = 3 y − 2 ⟹ x ( y − 1 ) = 3 y − 2 ⟹ x = y − 1 3 y − 2 .
The range is all real numbers except for y = 1 . Therefore, the range is R \ { 1 } .
Question 23 Solution Question 23: The y -intercept of y = a x 3 + b x 2 + c x + d is The y -intercept occurs when x = 0 . Substituting x = 0 into the equation, we get y = a ( 0 ) 3 + b ( 0 ) 2 + c ( 0 ) + d = d . The y -intercept is the point ( 0 , d ) , which can be written as ( 0 , y ( 0 )) .
Question 24 Solution Question 24: Evaluate lim h → 0 h 1 ln ( 2 2 + h ) .
We can rewrite the limit as lim h → 0 h l n ( 2 + h ) − l n ( 2 ) . This is the derivative of ln ( x ) evaluated at x = 2 . The derivative of ln ( x ) is x 1 , so the limit is 2 1 .
Question 25 Solution Question 25: Evaluate the left-hand limit lim x → − 3 − ( − x + 3 5 ) .
As x approaches − 3 from the left, x + 3 approaches 0 from the left, meaning x + 3 is a small negative number. Thus, x + 3 1 is a large negative number, and − x + 3 5 is a large positive number. Therefore, the limit is ∞ .
Question 26 Solution Question 26: Evaluate lim x → − 2 l n ( x + 3 ) x + 2 .
As x approaches − 2 , x + 2 approaches 0, and ln ( x + 3 ) approaches ln ( − 2 + 3 ) = ln ( 1 ) = 0 . This is an indeterminate form of type 0 0 , so we can use L'Hopital's rule. The derivative of x + 2 is 1, and the derivative of ln ( x + 3 ) is x + 3 1 . Thus, the limit is lim x → − 2 x + 3 1 1 = lim x → − 2 ( x + 3 ) = − 2 + 3 = 1 .
Question 27 Solution Question 27: A function f ( x ) is not continuous at a if ⋯ A function f ( x ) is not continuous at a if f ( a ) is undefined, or if lim x → a f ( x ) does not exist, or if lim x → a f ( x ) = f ( a ) . The most direct answer is that f ( a ) is undefined.
Final Answers Final Answers:
D. R \ { 1 }
D. ( 0 , y ( 0 ))
C. 2 1
A. − ∞ is incorrect. The correct answer is ∞ . Since this option is not available, we will assume there was a typo and the answer is A. ∞
D. 1
D. f ( a ) is undefined.
Examples
These types of questions are fundamental in calculus and real analysis. For example, understanding limits is crucial in physics for calculating instantaneous velocity and acceleration. Determining the range of a function is essential in economics for modeling supply and demand curves. Continuity is vital in engineering for designing stable systems.
