Determine the following limits.

lim x→−∞ (x+ √x^2−5x)

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Consider the expression \( x + \sqrt{x^2 - 5x} \) and factor out \( x \) from the square root to simplify.

Rewrite the expression as \( x + \sqrt{x^2(1 - \frac{5}{x})} \).

Simplify the square root to get \( x + |x|\sqrt{1 - \frac{5}{x}} \).

Since \( x \to -\infty \), \( |x| = -x \). Substitute this into the expression to get \( x - x\sqrt{1 - \frac{5}{x}} \).

Factor out \( x \) to simplify further: \( x(1 - \sqrt{1 - \frac{5}{x}}) \). Evaluate the limit as \( x \to -\infty \).

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Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior at points where it may not be explicitly defined, such as at infinity or discontinuities. In this case, we are interested in the limit as x approaches negative infinity.

Square Root Function

The square root function, denoted as √x, is a mathematical function that returns the non-negative value whose square is x. When dealing with limits involving square roots, it is essential to consider the behavior of the expression under the square root, especially as x approaches extreme values like negative infinity, which can affect the overall limit.

Dominant Terms in Polynomials

In polynomial expressions, the dominant term is the term with the highest degree, which significantly influences the behavior of the polynomial as x approaches infinity or negative infinity. For the limit in question, identifying the dominant terms in the expression x + √(x² - 5x) will help simplify the limit calculation and determine the overall behavior of the function.

Determine the following limits. lim x→−∞ (x+ √x^2−5x)