Question

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.95. lim(x→∞) (√(x² + x + 1) - √(x² - x))

Accepted Answer

Identify the limit expression: $$\lim_{x 	o \infty} \left( \sqrt{x$$^{2}$$ + x + 1} - \sqrt{x$$^{2}$$ - x} ight). $$Recognize that as $$x 	o \infty$$, both $$\sqrt{x$$^{2}$$ + x + 1}$$ and $$\sqrt{x$$^{2}$$ - x}$$ behave like $$x$$, so the expression is of the indeterminate form $$\infty - \infty. To $$apply l’Hôpital’s Rule, first rewrite the expression to a quotient form that yields an indeterminate form $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}. $$Multiply and divide by the conjugate to rationalize the expression: multiply numerator and denominator by $$\sqrt{x$$^{2}$$ + x + 1} + \sqrt{x$$^{2}$$ - x}. $$Simplify the numerator using the difference of squares formula: $$(a - b)(a + b) = a$$^{2}$$ - b$$^{2}$$, which gives $$\left( $$x^{2} + x + 1$$ ight) - \left( $$x^{2} - x$$ ight) = 2x + 1$$. Rewrite the limit as $$\lim_{x 	o \infty} \frac{2x + 1}{\sqrt{$$x^{2} + x + 1$$} + \sqrt{$$x^{2} - x$$}}$$ and then analyze this new expression to find the limit, possibly applying l’Hôpital’s Rule if the form is still indeterminate.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
95. lim(x→∞) (√(x² + x + 1) - √(x² - x))

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Key Concepts

Limits at Infinity

l’Hôpital’s Rule

Algebraic Manipulation of Radicals