Question

How do you find the asymptotes for $f\left( x \right) = \tan x$?

Answer 1

Hint: An asymptote is a line that touches the curve at infinity or we can say that it is a line such that the distance between the line and the curve approaches zero if the coordinates tend to infinity there will be two asymptotes of the given function one is horizontal and the other one is vertical. When we take a denominator equal to zero, we will get vertical asymptotes and if we take the $\mathop {\lim }\limits_{x \to \infty } $ for the function then we get horizontal asymptotes.

Complete step by step solution:
We have been given the function $f\left( x \right) = \tan x$.
We know that there will be two asymptotes one is horizontal and the other one is vertical. By denominator is equal to zero, we get the vertical asymptotes and if we take $x \to \infty $ limit on the function, we get horizontal asymptotes.
There will be no horizontal asymptotes for the tangent function as it increases and decreases without a bound between the vertical asymptotes. There is no break in the function from going maximum value to minimum value.
Now change the function in the form of sin and cos. Then,
$ \Rightarrow f\left( x \right) = \dfrac{{\sin x}}{{\cos x}}$
For the vertical asymptotes, we have,
$ \Rightarrow \cos x = 0$
The general solution of the above equation is $\left( {2n + 1} \right)\dfrac{\pi }{2}$.

Hence, the vertical asymptote is $\left( {2n + 1} \right)\dfrac{\pi }{2}$.

Note: Whenever we face such types of questions the key concept, we have to remember is that vertical asymptotes are the solution in terms of x and the horizontal asymptotes are the solution in terms of y, so use the definition of vertical and horizontal asymptotes which is stated above and solve for x and y, we will get the required vertical and horizontal asymptotes.