If \[\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \dfrac{a}{x} + \dfrac{b}{{{x^2}}}} \right)^{2x}} = {e^2}\] , then the values of \[a\] and \[b\] , are

Hint: In order to solve the above question, you need to know how to evaluate limits. Various methods are used in evaluating limits such as: factoring, conjugate, L’Hospital’s rule or certain formulas can be used. In the above question we will use a formula to simply the question and reduce the amount of time required to evaluate it.

Formula used:
The formula that will be used to evaluate the above limit is: \[\mathop {\lim }\limits_{x \to \infty } {\left( {1 + x} \right)^{\dfrac{1}{x}}} = e\] .

Complete step by step solution:
In the above question, we are given that \[\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \dfrac{a}{x} + \dfrac{b}{{{x^2}}}} \right)^{2x}} = {e^2}\] , with the help of this limit we will find the values of \[a\] and \[b\] .
Now we have, \[\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \dfrac{a}{x} + \dfrac{b}{{{x^2}}}} \right)^{2x}}\]
We will multiply and divide the power of the above limit by \[\left( {\dfrac{{\dfrac{a}{x} + \dfrac{b}{{{x^2}}}}}{{\dfrac{a}{x} + \dfrac{b}{{{x^2}}}}}} \right)\] ,
\[\mathop { = \lim }\limits_{x \to \infty } {\left( {1 + \dfrac{a}{x} + \dfrac{b}{{{x^2}}}} \right)^{2x\left( {\dfrac{{\dfrac{a}{x} + \dfrac{b}{{{x^2}}}}}{{\dfrac{a}{x} + \dfrac{b}{{{x^2}}}}}} \right)}}\] ,
Now, we will use the formula mentioned above, \[\mathop {\lim }\limits_{x \to \infty } {\left( {1 + x} \right)^{\dfrac{1}{x}}} = e\] ,we will get,
\[
   = {e^{\mathop {\lim }\limits_{x \to \infty } 2x\left( {\dfrac{a}{x} + \dfrac{b}{{{x^2}}}} \right)}} \\
   = {e^{2a}} \\
 \] .
But we also know that \[\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \dfrac{a}{x} + \dfrac{b}{{{x^2}}}} \right)^{2x}} = {e^2}\] ,
Therefore, we will now equate the values of \[\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \dfrac{a}{x} + \dfrac{b}{{{x^2}}}} \right)^{2x}}\] , we get,
\[ \Rightarrow {e^{2a}} = {e^2}\] ,
Now, we know if the bases are same then the power will be equal,
So,
\[
   \Rightarrow 2a = 2 \\
   \Rightarrow a = 1 \\
 \] .
From this we derive the value of \[a = 1\] .
And we can clearly see that \[b \in R\] .

Hence, the final answer is the value of \[a = 1\] and \[b \in R\] .

Note: To solve such questions involving limits, you must always keep in mind different methods that can be used to evaluate these questions. You must also remember different formulas that are used to simplify these questions. In the above question, with the help of a simple formula we were able to evaluate the answer of the question in a very less amount of time.