If f is a continuous function with even symmetry and lim x→∞ f(x)=10, which of the following statements must be true?
I. lim x→∞ f(x)=10
II. there are no vertical asymptotes
III. The lines y=10 and y= -10 are horizontal asymptotes
a) I only
b) II only
c) I and II only
d) All statements are true
The correct answer is c) I and II only.
Explanation:
I. Since the function f has even symmetry, it is symmetric about the y-axis. This means that as x approaches positive infinity, f(x) will approach the same value as when x approaches negative infinity. Therefore, lim x→∞ f(x) = 10.
II. Since the function f is continuous and has even symmetry, it has no vertical asymptotes. Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a certain value. However, in this case, as x approaches positive or negative infinity, f(x) approaches a constant value (10) and does not go to infinity. Hence, there are no vertical asymptotes.
III. The presence of even symmetry does not imply that the lines y=10 and y= -10 are horizontal asymptotes. Horizontal asymptotes occur when the function approaches a constant value as x approaches positive or negative infinity. In this case, since lim x→∞ f(x) = 10, the line y=10 is a horizontal asymptote. However, there is no information given about the behavior of f(x) as x approaches negative infinity. Therefore, we cannot conclude that the line y= -10 is a horizontal asymptote.
Hence, the correct statements are I and II only.
To determine which of the statements must be true, let's analyze each statement one by one.
Statement I: lim x→∞ f(x) = 10
Since the question states that the limit of f(x) as x approaches infinity is 10, this means that as x becomes very large, f(x) approaches 10. Since f(x) is continuous with even symmetry, this implies that as x becomes very large in the positive direction, f(x) also approaches 10. Therefore, Statement I is true.
Statement II: There are no vertical asymptotes
A vertical asymptote occurs when the function approaches either positive or negative infinity as x approaches a specific value. However, since f(x) is continuous with even symmetry, it means that it approaches the same value from both the positive and negative sides as x approaches any specific value. Therefore, there are no vertical asymptotes. Hence, Statement II is true.
Statement III: The lines y = 10 and y = -10 are horizontal asymptotes
Horizontal asymptotes occur when the function approaches a specific y-value as x approaches positive or negative infinity. In this case, since the limit of f(x) as x approaches infinity is 10, it means that the line y = 10 is a horizontal asymptote. However, with even symmetry, the same argument applies when x approaches negative infinity, indicating that y = -10 is also a horizontal asymptote. Therefore, both y = 10 and y = -10 are horizontal asymptotes. Hence, Statement III is true.
In conclusion, all three statements are true. Therefore, the correct answer is (d) All statements are true.
certainly (I), since that was given
not (II) -- consider y=10 + 1/x
not (III) since f(x) is even, so f(-x) = f(x)
III is true if f(x) is odd.