The function has an vertical asymptote of the function at option (A) x = -pi is the correct answer.
What is an asymptote?
An asymptote of the curve y = f(x) or in the implicit form f(x,y) = 0 is a straight line such that the distance between the curve and the straight line lends to zero when the points on the curve approach infinity.
For the given situation,
The graph below shows the function y = csc(x) and the asymptote.
The equation is y = csc(x).
⇒ [tex]f(x)=csc(x)[/tex]
⇒ [tex]f(x)=\frac{1}{sin(x)}[/tex]
When x approaches some constant value c from left or right, the curve moves towards infinity (i.e.,∞) , or -infinity (i.e., -∞) and this is called vertical asymptote.
Substitute x = 0,
⇒ [tex]f(0)=\frac{1}{sin(0)}[/tex]
⇒ [tex]f(0)=\frac{1}{0}[/tex]
Thus the asymptote of the function lies for x = 0, π, -π, 2π, -2π,...., nπ, -nπ values sin(x) will be 0. This implies, that csc(x) moves towards infinity, when x is the multiple of π or -π.
Hence we can conclude that the function has an vertical asymptote of the function at option (A) x = -pi is the correct answer.
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