Find the Asymptotes f(x)=(7sin(x))/(sin(x)+1)
The question asks to determine the asymptotes of the function f(x) = (7sin(x))/(sin(x)+1). Asymptotes are lines that the graph of a function approaches but never actually reaches as the independent variable (in this case x) heads towards positive or negative infinity or to a specific finite value. The problem involves analyzing the behavior of the trigonometric function as the values of x increase or decrease without bound or approach certain critical points where the denominator could be zero, leading to vertical asymptotes. It may also involve exploring the end behavior of the function to determine if there are any horizontal or oblique asymptotes.
$f \left(\right. x \left.\right) = \frac{7 sin \left(\right. x \left.\right)}{sin \left(\right. x \left.\right) + 1}$
Answer
Solution:
Step:1 The function \( f(x) = \frac{7\sin(x)}{\sin(x) + 1} \) does not exhibit any asymptotic behavior since the sine function is bounded and oscillatory, and does not tend towards infinity. Therefore, the function has no asymptotes.
Knowledge Notes:
Asymptotes are lines that a graph of a function approaches as the independent variable (usually \( x \)) either goes to infinity or to a specific point where the function is undefined. There are three types of asymptotes: horizontal, vertical, and oblique.
Horizontal Asymptotes: These occur if the function approaches a constant value as \( x \) approaches infinity or negative infinity. For rational functions, a horizontal asymptote is determined by comparing the degrees of the numerator and denominator polynomials.
Vertical Asymptotes: These occur at values of \( x \) where the function becomes unbounded or undefined. For rational functions, vertical asymptotes are found where the denominator is zero (and the numerator is not zero at the same point).
Oblique Asymptotes: These occur when the function approaches a line that is not horizontal as \( x \) goes to infinity or negative infinity. This usually happens when the degree of the numerator is exactly one greater than the degree of the denominator in a rational function.
For the given function \( f(x) = \frac{7\sin(x)}{\sin(x) + 1} \), the sine function \( \sin(x) \) oscillates between -1 and 1 for all real numbers \( x \). Therefore, the denominator \( \sin(x) + 1 \) oscillates between 0 and 2. Since the sine function is continuous and bounded, the function \( f(x) \) does not approach infinity or negative infinity as \( x \) increases or decreases without bound. Consequently, the function does not have any horizontal or vertical asymptotes.
Additionally, since the function is not a rational function where the degree of the numerator is one greater than the degree of the denominator, there are no oblique asymptotes either.
