Question

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

Answer 1

Hint: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,
\[ \Rightarrow \mathop {\lim }\limits_{x \to a} f(x) = \mathop {\lim }\limits_{h \to 0} f\left( {a + h} \right)\]