Question

$\lim_{x \to \infty} \left( \sqrt{x^2 + 5x - 7} - x \right) =$

• A. $\frac{7}{2}$
• B. 5
• C. $\frac{5}{2}$
• D. 6

Hint:

For limits of the form $\lim_{x \to \infty} (\sqrt{x^2 + bx + c} - x)$, there is a beautiful shortcut formula: the result is always exactly equal to $\frac{b}{2}$! Here, $b = 5$, so the answer is instantly $\frac{5}{2}$. This shortcut saves an incredible amount of scratchpad work!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem presents an algebraic limit as $x$ approaches infinity, which initially yields an indeterminate form of the type $\infty - \infty$. We need to rationalize the expression to evaluate its value.

Step 2: Key Formula or Approach:
To resolve limits involving an $\infty - \infty$ radical form, we multiply and divide by its conjugate expression to eliminate the troublesome difference: $$ \text{Conjugate} = \sqrt{x^2 + 5x - 7} + x $$

Step 3: Detailed Explanation:
Let's apply the rationalization technique to our limit expression: $$ \lim_{x \to \infty} \frac{\left(\sqrt{x^2 + 5x - 7} - x\right)\left(\sqrt{x^2 + 5x - 7} + x\right)}{\sqrt{x^2 + 5x - 7} + x} $$ Expanding the numerator using the difference of squares identity $(a-b)(a+b) = a^2 - b^2$: $$ = \lim_{x \to \infty} \frac{(x^2 + 5x - 7) - x^2}{\sqrt{x^2 + 5x - 7} + x} $$ $$ = \lim_{x \to \infty} \frac{5x - 7}{\sqrt{x^2(1 + \frac{5}{x} - \frac{7}{x^2})} + x} $$ Factor out $x$ from the square root in the denominator: $$ = \lim_{x \to \infty} \frac{5x - 7}{x\sqrt{1 + \frac{5}{x} - \frac{7}{x^2}} + x} = \lim_{x \to \infty} \frac{5x - 7}{x \left(\sqrt{1 + \frac{5}{x} - \frac{7}{x^2}} + 1\right)} $$ Divide every term in both the numerator and denominator by $x$: $$ = \lim_{x \to \infty} \frac{5 - \frac{7}{x}}{\sqrt{1 + \frac{5}{x} - \frac{7}{x^2}} + 1} $$ As $x \to \infty$, any fractional term with $x$ in its denominator tends to zero ($\frac{1}{x} \to 0$): $$ = \frac{5 - 0}{\sqrt{1 + 0 - 0} + 1} = \frac{5}{1 + 1} = \frac{5}{2} $$

Step 4: Final Answer:
The value of the limit is $\frac{5}{2}$, which corresponds to option (C).