Question

What is the limit of $\tan \left( {\dfrac{1}{x}} \right)$ as x approaches infinity?

Answer 1

In the above question we have to find the limit of $\tan \left( {\dfrac{1}{x}} \right)$ when x approaches infinity. Mathematically, we can also write it as $\mathop {\lim }\limits_{x \to \infty } \tan \left( {\dfrac{1}{x}} \right)$ . Now, to find the value of $\mathop {\lim }\limits_{x \to \infty } \tan \left( {\dfrac{1}{x}} \right)$ , we consider $\dfrac{1}{x}$ . Here, after finding the limit of $\dfrac{1}{x}$ when x tends to infinity, we can find the limit of $\tan \left( {\dfrac{1}{x}} \right)$ when x tends to infinity simply by putting $x \to \infty$.

Complete step by step solution:
Given function is $\tan \left( {\dfrac{1}{x}} \right)$
We have to find the limit of $\tan \left( {\dfrac{1}{x}} \right)$ when x tends to infinity.
Or, we have to find the value of
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \tan \left( {\dfrac{1}{x}} \right)$
Now consider $\mathop {\lim }\limits_{x \to \infty } \dfrac{1}{x}$ ,
When $x$ will approach infinity, then $\dfrac{1}{x}$ will approach zero.
Or we can write it as,
When $x \to \infty$ , then $\dfrac{1}{x} \to 0$ ...(1)
Now, similarly in $\mathop {\lim }\limits_{x \to \infty } \tan \left( {\dfrac{1}{x}} \right)$
When $\dfrac{1}{x}$ will approach zero, then $\tan \left( {\dfrac{1}{x}} \right)$ will also approach zero.
Because $\tan \left( 0 \right) = 0$
Therefore, we can also write it as
When $\dfrac{1}{x} \to 0$ then $\tan \left( {\dfrac{1}{x}} \right) \to 0$
Using ...(1) , we can write it as
When $x \to \infty$ then $\tan \left( {\dfrac{1}{x}} \right) \to 0$
Hence, now we can write the limit of $\tan \left( {\dfrac{1}{x}} \right) \to 0$ as
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \tan \left( {\dfrac{1}{x}} \right) = 0$
Therefore, the limit of $\tan \left( {\dfrac{1}{x}} \right)$ as x approaches infinity is 0.

Note:
When y = f(x) is a function of x and if at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. If these values tend to some definite unique number as x tends to a, then that obtained unique number is called the limit of f(x) at x = a.
Also, if we have a limit where x tends to a real number a then we can replace the limit $x \to a$ by a new limit as $h \to 0$ after putting $x = a + h$ in the given function.
Mathematically,
$\Rightarrow \mathop {\lim }\limits_{x \to a} f(x) = \mathop {\lim }\limits_{h \to 0} f\left( {a + h} \right)$