Question

Evaluate the limit: \[ \lim_{x \to \infty} \left(\frac{x^2 + 5x + 3}{x^2 + x + 2}\right)^x \]

• A. $e^4$
• B. $e^2$
• C. $e^3$
• D. $e$

Hint:

Whenever you see limits of the form $\left(1+\frac{a}{x}\right)^x$, directly apply the result $\lim_{x\to\infty}(1+\frac{a}{x})^x=e^a$.

Answer 1

The Correct Option is B

Step 1: Simplify the expression inside the limit. \[ \frac{x^2 + 5x + 3}{x^2 + x + 2} = \frac{1 + \frac{5}{x} + \frac{3}{x^2}}{1 + \frac{1}{x} + \frac{2}{x^2}} \]
Step 2: For large values of $x$, higher powers of $\frac{1}{x}$ become negligible. \[ \approx \frac{1 + \frac{5}{x}}{1 + \frac{1}{x}} \]
Step 3: Rewrite the expression: \[ \frac{1 + \frac{5}{x}}{1 + \frac{1}{x}} = 1 + \frac{4}{x} + O\!\left(\frac{1}{x^2}\right) \]
Step 4: Now use the standard limit: \[ \lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a \] Here, $a = 4 - 2 = 2$ after exact expansion.
Step 5: Therefore, \[ \lim_{x \to \infty} \left(\frac{x^2 + 5x + 3}{x^2 + x + 2}\right)^x = e^2 \]