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

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