Question

How can I solve that \[\dfrac{x}{{{\left( {{x}^{2}}+1 \right)}^{\dfrac{1}{2}}}}\] as \[x\] goes infinity?

Answer 1

Hint: From the question we have been asked to find the solution of the function when the x goes to infinity. So, to solve the questions of this type we will simplify the given function and reduce it into easier form which helps us to get solutions more accurately. Then we use the limits concept in mathematics and solve the given question.

Complete step by step solution:
Firstly, for the given question \[\dfrac{x}{{{\left( {{x}^{2}}+1 \right)}^{\dfrac{1}{2}}}}\] we will try and reduce the function into much simpler form. So , we get,
We will write the numerator in the given question in the terms of power of half as in the denominator of the function. So , we get,
\[\Rightarrow \dfrac{x}{{{\left( {{x}^{2}}+1 \right)}^{\dfrac{1}{2}}}}\]
\[\Rightarrow \dfrac{{{({{x}^{2}})}^{\dfrac{1}{2}}}}{{{\left( {{x}^{2}}+1 \right)}^{\dfrac{1}{2}}}}\]
Now, we will bring both the numerator and denominator into the same bracket. So, we get,
\[\Rightarrow {{\left( \dfrac{{{x}^{2}}}{{{x}^{2}}+1} \right)}^{\dfrac{1}{2}}}\]
Now, we will divide with \[{{x}^{2}}\] in the both numerator and denominator for the above expression. We get,
\[\Rightarrow {{\left( \dfrac{\dfrac{{{x}^{2}}}{{{x}^{2}}}}{\dfrac{{{x}^{2}}+1}{{{x}^{2}}}} \right)}^{\dfrac{1}{2}}}\]
\[\Rightarrow {{\left( \dfrac{1}{1+\dfrac{1}{{{x}^{2}}}} \right)}^{\dfrac{1}{2}}}\]
We know the formula in limits which is \[\displaystyle \lim_{x \to \infty }\left( \dfrac{1}{{{x}^{2}}} \right)=0\]. So , we get the equation reduced when we use this basic formula in limits as follows.
\[\Rightarrow 1\]

Therefore, the solution to the given question is \[1\].

Note: We must be very careful in doing the calculations. We must have good knowledge in the concept of limits. We must be able to understand this kind of question and can be able to do the simplifications of the functions so that limits can be easily applied. We must know the formula \[\displaystyle \lim_{x \to \infty }\left( \dfrac{1}{{{x}^{2}}} \right)=0\] to solve this question.