`lim_(nto oo) (2^n+5^n)^(1//n)` is equal to

Options

A. 2
B. 5
C. e
D. none of these

Explanation

To solve the limit \( \lim_{n \to \infty} (2^n + 5^n)^{1/n} \), we can follow these steps:

### Step 1: Factor out the dominant term
The term \( 5^n \) grows faster than \( 2^n \) as \( n \) approaches infinity. Therefore, we can factor \( 5^n \) out of the expression inside the limit.

\[ (2^n + 5^n)^{1/n} = \left(5^n\left(\frac{2^n}{5^n} + 1\right)\right)^{1/n} \]

### Step 2: Simplify the expression
Now we can simplify the expression:

\[ = \left(5^n\right)^{1/n} \left(\frac{2^n}{5^n} + 1\right)^{1/n} \]

### Step 3: Evaluate the first part
The first part simplifies as follows:

\[ \left(5^n\right)^{1/n} = 5 \]

### Step 4: Evaluate the second part
Now we need to evaluate \( \left(\frac{2^n}{5^n} + 1\right)^{1/n} \):

\[ \frac{2^n}{5^n} = \left(\frac{2}{5}\right)^n \]

As \( n \to \infty \), \( \left(\frac{2}{5}\right)^n \) approaches 0 since \( \frac{2}{5} < 1 \). Therefore:

\[ \left(\frac{2^n}{5^n} + 1\right)^{1/n} = \left(0 + 1\right)^{1/n} = 1^{1/n} = 1 \]

### Step 5: Combine the results
Now we combine the results from Steps 3 and 4:

\[ \lim_{n \to \infty} (2^n + 5^n)^{1/n} = 5 \cdot 1 = 5 \]

### Final Result
Thus, the limit is:

\[ \lim_{n \to \infty} (2^n + 5^n)^{1/n} = 5 \]