6. Find the exact value of [tex]$\lim _{x \rightarrow 0} \frac{x}{x}$[/tex].
A. [tex]$1 / \pi$[/tex]
B. 0
C. 1
7. Which of the following functions is continuous?
A. [tex]$f(x)=|x|$\qquad$[/tex] C. [tex]$f(x)=\frac{1}{x}$[/tex]
B. [tex]$f(x)=\left{\begin{array}{ll}3 & text { if } x<4, \\ \frac{x}{2}+3 & text { if } x \geq 4,\end{array}\right.$[/tex]
(D) [tex]$f(x)=\left{\begin{array}{ll}\ln (x) & text { if } x<0, \\ 0 & text { if } x=0,\end{array}\right.$[/tex]
8. If [tex]$[x]$[/tex] is the greatest integer not greater than [tex]$x$[/tex], then [tex]$\lim _{x \rightarrow 1 / 2}[x]$[/tex].
A. [tex]$1 / 2$[/tex]
B. 1
C. 0
D. None of them
9. Let [tex]$f(x)=\left{\begin{array}{lll}x^2 & 1 & text { if } x \neq 1 . \\ 4 & text { if } x=1 .\end{array}\right.$[/tex]
Which of the following statements is/are true?
I. [tex]$\lim _{x \rightarrow 1} f(x)$[/tex] exists
II. [tex]$f(1)$[/tex] exists
III. [tex]$f$[/tex] is continuous at [tex]$x=1$[/tex].
A. only I
B. only II
C. I and II
D. None of them
10. Find the acute angle between two straight lines having the equations
Answer (2)
The answers to the questions are: 6. C (1), 7. A ( ∣ x ∣ ), 8. C (0), 9. C (I and II), and 10 (assuming example lines) is 45 degrees. Each question covers important concepts of limits, continuity, and angles in mathematics.
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Question 6: The limit of x x as x approaches 0 is 1 .
Question 7: The continuous function is f ( x ) = ∣ x ∣ , option A .
Question 8: The limit of [ x ] as x approaches 2 1 is 0 , option C.
Question 9: Statements I and II are true, option C .
Question 10: The acute angle between the example lines y = x and y = 0 is 45 degrees.
Explanation
Introduction We will solve questions 6 through 10, which cover limits and continuity.
Question 6: Limit of x/x For question 6, we need to find the limit of x x as x approaches 0. Since x x = 1 for all x = 0 , the limit as x approaches 0 is 1.
Question 7: Continuity of Functions For question 7, we need to determine which of the given functions is continuous. A function is continuous if it is defined at a point, the limit exists at that point, and the limit equals the function value. Let's analyze each option:
A. f ( x ) = ∣ x ∣ is continuous everywhere. B. f ( x ) = { 3 2 x + 3 if x < 4 if x ≥ 4 . We need to check continuity at x = 4 . As x approaches 4 from the left, f ( x ) = 3 . As x approaches 4 from the right, f ( x ) = 2 4 + 3 = 5 . Since the left and right limits are not equal, this function is discontinuous at x = 4 .
C. f ( x ) = x 1 is discontinuous at x = 0 .
D. f ( x ) = { ln ( x ) 0 if x < 0 if x = 0 . The natural logarithm ln ( x ) is only defined for 0"> x > 0 , so this function is not defined for x < 0 . Thus, this function is not continuous. Therefore, the continuous function is f ( x ) = ∣ x ∣ .
Question 8: Limit of Greatest Integer Function For question 8, we need to find the limit of the greatest integer function [ x ] as x approaches 2 1 . The greatest integer function [ x ] returns the largest integer less than or equal to x . As x approaches 2 1 , the greatest integer less than or equal to x is 0.
Question 9: Analysis of Piecewise Function For question 9, we have the function f ( x ) = { x 2 4 if x = 1 if x = 1 . We need to determine which statements are true:
I. lim x → 1 f ( x ) exists. As x approaches 1, f ( x ) = x 2 , so lim x → 1 f ( x ) = 1 2 = 1 . Thus, the limit exists. II. f ( 1 ) exists. f ( 1 ) = 4 , so f ( 1 ) exists. III. f is continuous at x = 1 . For f to be continuous at x = 1 , we need lim x → 1 f ( x ) = f ( 1 ) . We have lim x → 1 f ( x ) = 1 and f ( 1 ) = 4 . Since 1 = 4 , f is not continuous at x = 1 . Therefore, only statements I and II are true.
Question 10: Angle Between Two Lines For question 10, we need to find the acute angle between two straight lines. Since the equations of the lines are not provided, I will assume two example lines: y = x and y = 0 (x-axis). The slopes are m 1 = 1 and m 2 = 0 . The angle θ between the lines is given by tan ( θ ) = 1 + m 1 m 2 m 1 − m 2 = 1 + 1 ⋅ 0 1 − 0 = 1 . Therefore, θ = arctan ( 1 ) = 45 degrees.
Examples
Limits and continuity are fundamental concepts in calculus and are used in various real-world applications. For example, in physics, understanding continuity is crucial when modeling the motion of objects or the flow of fluids. In economics, continuity helps in analyzing supply and demand curves. In computer graphics, continuous functions are used to create smooth curves and surfaces. Understanding these concepts provides a foundation for more advanced mathematical modeling and problem-solving in these fields.
