To evaluate the limit [tex]\(\lim_{n \rightarrow \infty} \sqrt[n]{10} + 5\)[/tex], we should look at the behavior of the term [tex]\(\sqrt[n]{10}\)[/tex] as [tex]\(n\)[/tex] approaches infinity.
Given the limit [tex]\(\lim_{n \rightarrow \infty} \sqrt[n]{10} + 5\)[/tex], we can proceed as follows:
1. Understand the expression: The term [tex]\(\sqrt[n]{10}\)[/tex] can be rewritten as [tex]\(10^{\frac{1}{n}}\)[/tex].
2. Behavior of [tex]\(10^{\frac{1}{n}}\)[/tex]: As [tex]\(n\)[/tex] increases, the exponent [tex]\(\frac{1}{n}\)[/tex] approaches zero. Anything raised to the power of zero approaches one, thus [tex]\(10^{\frac{1}{n}}\)[/tex] approaches 1 as [tex]\(n\)[/tex] approaches infinity.
3. Formal limit analysis:
[tex]\[
\lim_{n \rightarrow \infty} 10^{\frac{1}{n}} = 1
\][/tex]
4. Add the constant: Once we have the limit of [tex]\(10^{\frac{1}{n}}\)[/tex] as 1, we can add the constant 5:
[tex]\[
\lim_{n \rightarrow \infty} \left(10^{\frac{1}{n}} + 5\right) = 1 + 5
\][/tex]
5. Compute the final limit:
[tex]\[
1 + 5 = 6
\][/tex]
Thus, the limit is:
[tex]\[
\lim_{n \rightarrow \infty} \sqrt[n]{10} + 5 = 6
\][/tex]
