Question

What is the limit of \[{{\left( 1+\dfrac{2}{x} \right)}^{x}}\] as \[x\] approaches \[\infty \]?

Answer 1

Hint: To find out the limit of the function as it approaches \[\infty \] is by just having a look at the degree of the function. Degree is nothing but the highest exponent in the function. If the degree of the function is greater than \[0\] then the limit is \[\infty \] or \[-\infty \] and if the degree of the function is less than \[0\], then the limit is \[0\].

Complete step-by-step solution:
Now let us try to learn more about limits approaching infinity.
So if we find that the limit is approaching positive infinity from one side and then it is reaching negative infinity from the other side then it does not approach the same thing from both sides. In this case, the limit does not exist.
Now, let us start solving the function \[{{\left( 1+\dfrac{2}{x} \right)}^{x}}\]
To solve this, let us use the general formula \[\displaystyle \lim_{u \to \infty }{{\left( 1+\dfrac{1}{u} \right)}^{u}}=e\]
We can write \[\dfrac{2}{x}\] as \[\dfrac{1}{\dfrac{x}{2}}\] , for our easy calculation and conversion.
From the above mentioned formula, we can extract that
\[{{\left( 1+\dfrac{2}{x} \right)}^{x}}\]\[={{\left( 1+\dfrac{1}{\dfrac{x}{2}} \right)}^{x}}\]
Now multiply and divide the whole function with 2.
We get,
\[\displaystyle \lim_{x \to \infty }{{\left( {{\left( 1+\dfrac{1}{\dfrac{x}{2}} \right)}^{\dfrac{x}{2}}} \right)}^{2}}\]
Now from the above mentioned formula , we can observe that \[u=\dfrac{x}{2}\].
\[\therefore \] On evaluating the limit we get,
\[{{e}^{2}}\]

Note: The common error could be the way of approach. Before solving this, we need to check out the limit to which the function is approaching and solve accordingly. We must note that limits do not exist when the function doesn’t approach a particular value or a finite value.