f(x) = tan x / sin x
Find the vertical asymptote.
Describe its behavior to the left and right of the vertical asymptote.
tanx/sinx = 1/cosx
cosx=0 at odd multiples of pi/2.
So, that is where you have asymptotes.
cosx changes sign at those points, so also does your graph.
Well, I must say, finding the vertical asymptote in this case is quite the juggling act! To find it, we need to determine when the denominator becomes zero. Since the denominator is sin x, it will be zero whenever x is equal to kπ, where k is an integer.
Now, when it comes to the behavior of our function to the left and right of the vertical asymptote, things get pretty entertaining. As x approaches a value where the denominator becomes zero, our function f(x) will start clowning around and seem to blow up to infinity or negative infinity. So, you could say it gets quite dramatic!
To put it simply, to the left of the vertical asymptote, the function will approach either positive or negative infinity, depending on which side you're coming from. And to the right of the vertical asymptote, it will do the same, but in the opposite direction. It's like a never-ending circus act!
Just remember, while our function may seem wild near the vertical asymptote, it's all part of the mathematical circus! Keep enjoying the show and work those equations with a smile on your face!
To find the vertical asymptote of the function f(x) = tan(x) / sin(x), we need to determine the values of x for which the function approaches infinity or negative infinity.
We know that tan(x) is undefined at odd multiples of π/2, where the tangent function has vertical asymptotes.
First, let's find the values of x for which sin(x) = 0 (denominator equals zero). Since sin(x) = 0 when x = nπ, where n is an integer, we have:
sin(x) = 0
x = nπ
Therefore, our potential values for vertical asymptotes are x = nπ, where n is an integer.
Now, let's analyze the behavior of the function to the left and right of these potential vertical asymptotes.
1. Left of the vertical asymptotes (as x approaches π/2, 3π/2, 5π/2, etc.):
- In this region, sin(x) is positive, and tan(x) approaches positive or negative infinity depending on the specific value of n. Hence, f(x) approaches positive or negative infinity:
lim(x -> (nπ - ε)^+) f(x) = ±∞
2. Right of the vertical asymptotes (as x approaches -π/2, -3π/2, -5π/2, etc.):
- In this region, sin(x) is negative, and tan(x) approaches positive or negative infinity depending on the specific value of n. Therefore, f(x) also approaches positive or negative infinity:
lim(x -> (nπ + ε)^-) f(x) = ±∞
To summarize, the vertical asymptotes of the function f(x) = tan(x) / sin(x) are x = nπ, and to the left and right of these asymptotes, f(x) approaches positive or negative infinity.
To find the vertical asymptote of the given function f(x) = tan x / sin x, we need to identify the values of x for which the function approaches positive or negative infinity.
Let's start by simplifying the function using the trigonometric identity tan x = sin x / cos x:
f(x) = tan x / sin x = (sin x / cos x) / sin x
Now, we can cancel out the sin x terms:
f(x) = 1 / cos x
Since cos x becomes zero at points where x = (2n + 1) * π / 2 (where n is an integer), these are the potential vertical asymptotes of the function.
To describe the behavior of the function to the left and right of the vertical asymptotes, we need to examine the sign of f(x) as x approaches these values.
1. Left of the vertical asymptote (approaching from the negative side):
As x approaches the left side of the vertical asymptote, the values of cos x become negative. For cos x < 0, 1 / cos x is negative. Therefore, the function approaches negative infinity as x approaches the vertical asymptote from the left.
2. Right of the vertical asymptote (approaching from the positive side):
As x approaches the right side of the vertical asymptote, the values of cos x become positive. For cos x > 0, 1 / cos x is positive. Therefore, the function approaches positive infinity as x approaches the vertical asymptote from the right.
To summarize:
- The vertical asymptotes of the function are at x = (2n + 1) * π / 2 (where n is an integer).
- The behavior of the function to the left of the vertical asymptote is approaching negative infinity.
- The behavior of the function to the right of the vertical asymptote is approaching positive infinity.