10) lim _(xarrow -infty )(x^2+xsqrt (x^2-1))=?

To find the limit of the given expression as \( x \) approaches negative infinity, we can use the following steps:<br /><br />1. Identify the dominant term in the expression.<br />2. Simplify the expression by focusing on the dominant term.<br /><br />Given expression: \( \lim_{x \to -\infty} (x^2 + x\sqrt{x^2 - 1}) \)<br /><br />### Step 1: Identify the dominant term<br />As \( x \) approaches negative infinity, the term \( x^2 \) will dominate all other terms in the expression because it grows faster than the other terms.<br /><br />### Step 2: Simplify the expression<br />We can factor out \( x^2 \) from the entire expression to isolate the dominant term:<br /><br />\[ \lim_{x \to -\infty} (x^2 + x\sqrt{x^2 - 1}) = \lim_{x \to -\infty} x^2 \left(1 + \frac{\sqrt{x^2 - 1}}{x}\right) \]<br /><br />Now, we need to evaluate the limit of the term inside the parentheses:<br /><br />\[ \lim_{x \to -\infty} \left(1 + \frac{\sqrt{x^2 - 1}}{x}\right) \]<br /><br />As \( x \) approaches negative infinity, \( x^2 - 1 \) becomes very large, and we can approximate \( \sqrt{x^2 - 1} \) using the binomial expansion for large \( |x| \):<br /><br />\[ \sqrt{x^2 - 1} \approx x\sqrt{1 - \frac{1}{x^2}} \]<br /><br />Thus,<br /><br />\[ \frac{\sqrt{x^2 - 1}}{x} \approx \frac{x\sqrt{1 - \frac{1}{x^2}}}{x} = \sqrt{1 - \frac{1}{x^2}} \]<br /><br />As \( x \) approaches negative infinity, \( \frac{1}{x^2} \) approaches 0, so:<br /><br />\[ \sqrt{1 - \frac{1}{x^2}} \approx 1 \]<br /><br />Therefore,<br /><br />\[ \lim_{x \to -\infty} \left(1 + \frac{\sqrt{x^2 - 1}}{x}\right) \approx \lim_{x \to -\infty} \left(1 + 1\right) = 2 \]<br /><br />### Final Step: Combine the results<br />Now, we multiply this result by the dominant term \( x^2 \):<br /><br />\[ \lim_{x \to -\infty} x^2 \cdot 2 = 2x^2 \]<br /><br />As \( x \) approaches negative infinity, \( 2x^2 \) approaches positive infinity. However, since the original expression involves \( x^2 \) plus another term, we must consider the behavior of the entire function.<br /><br />The correct interpretation is that the limit of the entire expression is dominated by the \( x^2 \) term, but since it is squared and multiplied by a constant, the result is finite and non-negative. Therefore, the limit is:<br /><br />\[ \boxed{+\infty} \]