Question

How do I find the limit as x approaches infinity of $\tan x$?

Answer 1

Given a trigonometric expression. We have to evaluate the limit of the expression, when x approaches infinity. The tangent function is a continuous function, therefore we will move the limit inside the trigonometric function. Then, apply the limit to x. If the leading coefficient is positive, then the limit at infinity of the polynomial is always infinity.

Complete step by step solution:
We are given the expression, $\tan x$.
Apply the limits to the expression.
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \left( {\tan x} \right)$
Since tangent is a continuous function, move the limit inside the expression.
$\Rightarrow \tan \left( {\mathop {\lim }\limits_{x \to \infty } x} \right)$
Now, apply the limits to the variable x.
$\Rightarrow \tan \left( \infty \right)$
Thus, the value of $\tan \left( \infty \right)$does not exist, which means $\tan x$ diverges as x approaches infinity.

Final answer: Hence, the value of $\tan \left( \infty \right)$does not exist as x approaches to infinity.

Additional Information:
The trigonometric functions are the functions which are generally used to find the coordinates of a particular point on any circle using the measure of an angle. There are six basic types of trigonometric functions. These are sine, cosine, tangent, secant, cosecant and cotangent functions. Any trigonometric function can be defined with the corresponding angle theta. These trigonometric functions are called periodic functions, such as the period of sine and cosine function is $2\pi$ because after this period the function repeats itself. Similarly, the period of tangent and cotangent function is $\pi$. The number of vertical asymptotes in the tangent function are infinite, therefore it does not approach a finite limit.

Note:
Please note that the tangent is a continuous function which oscillates between $- \infty$ to $\infty$. The range of $\tan \left( {x - \pi } \right)$ and $\tan \left( {x + \pi } \right)$ will include all values from negative infinity to positive infinity, for whatever value of x we will choose.