Certainly! Let's analyze the limit of the given expression as [tex]\( n \)[/tex] approaches infinity.
We are given the expression [tex]\(\lim_{n \to \infty}\left(\frac{2}{\sqrt{n}} + \frac{3}{\sqrt[3]{n}}\right)\)[/tex].
To solve this limit, we will consider each term separately and then combine the results.
First, let's analyze the term [tex]\(\frac{2}{\sqrt{n}}\)[/tex]:
1. As [tex]\( n \)[/tex] approaches infinity, [tex]\(\sqrt{n}\)[/tex] (the square root of [tex]\( n \)[/tex]) also approaches infinity, because larger values of [tex]\( n \)[/tex] result in larger square roots.
2. Therefore, as [tex]\(\sqrt{n}\)[/tex] grows very large, [tex]\(\frac{2}{\sqrt{n}}\)[/tex] becomes smaller and approaches 0. Explicitly, for a very large [tex]\( n \)[/tex], the fraction [tex]\(\frac{2}{\sqrt{n}}\)[/tex] is very close to 0.
Next, let's consider the term [tex]\(\frac{3}{\sqrt[3]{n}}\)[/tex]:
1. As [tex]\( n \)[/tex] approaches infinity, [tex]\(\sqrt[3]{n}\)[/tex] (the cube root of [tex]\( n \)[/tex]) also approaches infinity. Like with the square root, as [tex]\( n \)[/tex] gets larger, the cube root of [tex]\( n \)[/tex] increases.
2. Consequently, as [tex]\(\sqrt[3]{n}\)[/tex] becomes very large, [tex]\(\frac{3}{\sqrt[3]{n}}\)[/tex] similarly becomes smaller and approaches 0. Again, for very large values of [tex]\( n \)[/tex], the term [tex]\(\frac{3}{\sqrt[3]{n}}\)[/tex] is very close to 0.
Now, we combine these two observations:
Since both [tex]\(\frac{2}{\sqrt{n}}\)[/tex] and [tex]\(\frac{3}{\sqrt[3]{n}}\)[/tex] approach 0 as [tex]\( n \)[/tex] approaches infinity, their sum will also approach 0. Formally, when we sum two quantities that both go to 0 in the limit, their sum also goes to 0.
Therefore, we conclude:
[tex]\[
\lim_{n \to \infty} \left( \frac{2}{\sqrt{n}} + \frac{3}{\sqrt[3]{n}} \right) = 0
\][/tex]
By exploring a specific example where [tex]\( n \)[/tex] is extremely large (like [tex]\( n = 10^{10} \)[/tex]), we can demonstrate this:
1. For [tex]\( n = 10^{10} \)[/tex]:
- [tex]\(\frac{2}{\sqrt{10^{10}}}\)[/tex] approximately equals to [tex]\( 2 \times 10^{-5} = 2e-05 \)[/tex]
- [tex]\(\frac{3}{\sqrt[3]{10^{10}}}\)[/tex] approximately equals to [tex]\( 3 \times (10^{10})^{-1/3} \approx 0.0013924766500838343 \)[/tex]
2. Summing these values gives approximately [tex]\( 0.0014124766500838344 \)[/tex].
While these computed values are approaching zero, this specific number is just an intermediate step showing that both parts indeed get very small, supporting our general conclusion that:
[tex]\[
\lim_{n \to \infty} \left( \frac{2}{\sqrt{n}} + \frac{3}{\sqrt[3]{n}} \right) = 0
\][/tex]
Thus, we have shown that our limit converges to 0 as [tex]\( n \)[/tex] approaches infinity.
