Question

Let n represent any integer.

Answer 1

Answer:

Vertical asymptote of any function occur at all those points where the value of the function is not defined.

So, to find vertical asymptotes of the given function, we find all the values of x for which the given functions are not defined

A. tan x

tan x is not defined for x = π/2 and integer multiples of π

⇒ tan x has a vertical asymptote at x = π/2 + nπ , where n is any integer

B. cot x

cot x is not defined for x = π and since it has period π so it will be not defined for all integer multiples of π

⇒ cot x has a vertical asymptote at x = nπ , where n is any integer

C. sec x

sec x is not defined for x = π/2 and integer multiples of π

⇒ sec x has a vertical asymptote at x = π/2 + nπ , where n is any integer

D. csc x

csc x is not defined for x = π and since it has period π so it will be not defined for all integer multiples of π

⇒ csc x has a vertical asymptote at x = nπ , where n is any integer

Answer 2

Answer: Hello mate!

The asymptotes are located at the points where the denominator is equal to zero, then, let's solve all the cases:

A) We know that tan(x) = sen(x)/cos(x), so the asymptote is at the value of x that makes the cosine equal to zero.

This is pi/2

we know that tan(x) has an asymptote at pi/2, but because of the bheaviour of the trigonometric function, tan(x) also has an asimptote at pi/2 + pi.

then if we define n as an integer number, the asimpotes of tan(x) are when x = pi/2 + n*pi

B) now we ave cot(x) = 1/tan(x) = cos(x)/sin(x), so the asymptote is at the points where sin(x) is equal to zero, that are the points of the form x = pi + n*pi

C) Sec(x) = 1/cos(x), then the asympotes are, like in part A, at the values of x where cos(x) = 0, so the asympotetes are at the values of x = pi/2 + n*pi

D) cosec(x) = 1/sin(x), again, the asymptotes are when sin(x) = 0, so the asympotes are in the values of x = pi + n*pi

Where in all the situations n is an integer.