Question

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

Answer 1

Hint: This is a standard format of finding the limit of an indefinite form. Here is indefinite form which is forming is ${1^\infty }$. One to the power infinity is known as an indeterminate form, because it is unknown. One to the power infinity is unknown because infinity itself is endless.

Complete answer:
In the above question, we have to find the limit of ${\left( {\dfrac{x}{{x + 1}}} \right)^x}$.
We can also write it as,
$ \Rightarrow {\lim _{x \to \infty }}{\left( {\dfrac{x}{{x + 1}}} \right)^x}$
We can convert it into the format ${\lim _{x \to \infty }}{\left( {1 + f\left( x \right)} \right)^{g\left( x \right)}} = {e^{{{\lim }_{x \to \infty }}f\left( x \right)g\left( x \right)}}$
It is clear that here an indefinite form is forming as ${1^\infty }$.
First, we have to convert our given question in the required format. But for that we have to add and subtract $ - 1$.
$ \Rightarrow {\lim _{x \to \infty }}{\left( {1 + \dfrac{x}{{x + 1}} - 1} \right)^x}$
$ \Rightarrow {\lim _{x \to \infty }}{\left( {1 + \dfrac{{x - \left( {x + 1} \right)}}{{x + 1}}} \right)^x}$
Now on comparing above equation with ${\lim _{x \to \infty }}{\left( {1 + f\left( x \right)} \right)^{g\left( x \right)}} = {e^{{{\lim }_{x \to \infty }}f\left( x \right)g\left( x \right)}}$, we get
$ \Rightarrow {e^{{{\lim }_{x \to \infty }}\left( {\dfrac{{x - \left( {x + 1} \right)}}{{x + 1}}} \right) \times x}}$
$ \Rightarrow {e^{{{\lim }_{x \to \infty }}\left( {\dfrac{{ - 1}}{{x + 1}}} \right) \times x}}$
On simplification, we get
\[ \Rightarrow {e^{{{\lim }_{x \to \infty }}\left( {\dfrac{{ - x}}{{x + 1}}} \right)}}\]
Now, divide numerator and denominator by x.
\[ \Rightarrow {e^{{{\lim }_{x \to \infty }}\left( {\dfrac{{ - 1}}{{1 + \dfrac{1}{x}}}} \right)}}\]
Now on applying limits, we get
 \[ \Rightarrow {e^{ - 1}}\]
Therefore, the value of the above integral is \[{e^{ - 1}}\].

Note:
When would we encounter a situation like ${1^\infty }$. You want to find the limit of the following function as it approaches infinity. For example, ${\lim _{x \to \infty }} = {1^x}$. When we plug infinity into this function, we see that it takes on the indeterminate form of one to the power infinity.