Which of the following is an asymptote of f(x)=(x^2+3x+2)/(x+2)*tan(pi*x)?
A) x=-2
B) x=-1
C) x=1/2
D) x=1
E) x=2
Answer is C but I got A because substituting x for -2 makes the graph undefined.
This is a sneaky question,
f(x) = (x^2+3x+2)/(x+2)*tan(π*x)
= (x+1)(x+2)/(x+2) tan(πx)
= (x+1)tan(πx)
the vertical asymptote is caused by the tan(πx) part,
recall that tan90° is undefined, and 90° = π/2 radians or (1/2)π
so if x = 1/2 we have a vertical asymptote
when x = -2, we have a "hole" in our graph, which of course would not be visible in a physical graph
http://www.wolframalpha.com/input/?i=y+%3D+%28x%5E2%2B3x%2B2%29%2F%28x%2B2%29*tan%28%CF%80*x%29
To determine the asymptotes of the given function, f(x) = (x^2 + 3x + 2)/(x + 2) * tan(pi*x), we need to consider two types of asymptotes: vertical asymptotes and horizontal asymptotes.
Vertical Asymptotes:
The vertical asymptotes occur when the denominator of the function equals zero, since division by zero is undefined. In this case, the vertical asymptote occurs when x+2 = 0. Solving for x, we find x = -2. Therefore, option A (x = -2) is a vertical asymptote of the function.
Horizontal Asymptotes:
To find the horizontal asymptotes, we need to analyze the behavior of the function as x approaches positive and negative infinity.
Let's focus on the expression tan(pi*x) separately. This function oscillates infinitely as x goes towards positive and negative infinity. Therefore, it does not have any horizontal asymptotes.
However, we still need to consider the other part of the function, (x^2 + 3x + 2)/(x + 2). As x approaches positive or negative infinity, the highest order term in the numerator (x^2) will dominate the function.
Dividing both the numerator and denominator by x, we get the simplified form: (x + 3 + 2/x) / (1 + 2/x). Now as x approaches infinity, the term 2/x approaches zero, and the function approaches the value (x + 3) / 1, which is simply x + 3.
Therefore, as x approaches positive infinity, the function approaches the line y = x + 3, and as x approaches negative infinity, the function also approaches the line y = x + 3.
Hence, the function f(x) = (x^2 + 3x + 2)/(x + 2) * tan(pi*x) has two horizontal asymptotes given by the equations y = x + 3.
Since option C (x = 1/2) is not a vertical asymptote and the function does not have any horizontal asymptotes, the answer should not be C. It is recommended to review the question and the given options to ensure the correct answer.