Question

$\lim _{x \rightarrow \infty}\left(1+a^{2}\right)^{x} \cdot \frac{b}{\left(1+a^{2}\right)^{x}}\( is \)(a, b \in R)$ (A) \(\sqrt{b}\) (B) b (C) \(\mathrm{b}^{2}\) (D) none of these

Short answer

Question: Find the limit of the given function as \(x \rightarrow \infty\): \(\lim _{x \rightarrow \infty}\left((1+a^{2})^{x} \cdot \frac{b}{(1+a^{2})^{x}}\right)\) Options: A) \(\sqrt{b}\) B) b C) \({b}^{2}\) D) none of these Answer: (B) b

Step-by-step solution

Write down the given function

The given function is: \(\lim _{x \rightarrow \infty}\left((1+a^{2})^{x} \cdot \frac{b}{(1+a^{2})^{x}}\right)\)

Simplify the given function

We can simplify the given function by dividing the base of the power function in the numerator and denominator. \(\lim _{x \rightarrow \infty}\left(\frac{(1+a^{2})^{x} \cdot b}{(1+a^{2})^{x}}\right)\) By simplifying: \(\lim _{x \rightarrow \infty} \left(b\right)\)

Determine the limit

Since b is a constant, the limit is itself as x approaches infinity: \(\lim _{x \rightarrow \infty} \left(b\right) = b\)

Choose the correct option

By comparing the result to the given options, we see that our result matches option (B): Answer: (B) b