Questions: Calculate the limit. (Use symbolic notation and fractions where needed.) lim as x approaches infinity of (sqrt(x^4 + 11x) - x^2) =

Solution Steps

To solve the limit \(\lim _{x \rightarrow \infty}\left(\sqrt{x^{4}+11 x}-x^{2}\right)\), we can simplify the expression by rationalizing it. This involves multiplying the expression by its conjugate, which is \(\sqrt{x^{4}+11x} + x^2\), and then simplifying the resulting expression. This will help us to identify the behavior of the function as \(x\) approaches infinity.

We start with the limit expression:

\[ \lim _{x \rightarrow \infty}\left(\sqrt{x^{4}+11 x}-x^{2}\right) \]

To simplify, we rewrite the expression as:

\[ \sqrt{x^{4}+11 x} - x^{2} \]

Next, we multiply the expression by its conjugate:

\[ \frac{\left(\sqrt{x^{4}+11 x}-x^{2}\right)\left(\sqrt{x^{4}+11 x}+x^{2}\right)}{\sqrt{x^{4}+11 x}+x^{2}} \]

This results in:

\[ \frac{x^{4}+11 x - x^{4}}{\sqrt{x^{4}+11 x}+x^{2}} = \frac{11 x}{\sqrt{x^{4}+11 x}+x^{2}} \]

Now, we evaluate the limit as \(x\) approaches infinity:

\[ \lim_{x \to \infty} \frac{11 x}{\sqrt{x^{4}+11 x}+x^{2}} \]

As \(x\) approaches infinity, the dominant term in the square root is \(x^4\), so we can approximate:

\[ \sqrt{x^{4}+11 x} \approx x^{2} \]

Thus, the limit simplifies to:

\[ \lim_{x \to \infty} \frac{11 x}{x^{2} + x^{2}} = \lim_{x \to \infty} \frac{11 x}{2x^{2}} = \lim_{x \to \infty} \frac{11}{2x} = 0 \]

Final Answer