Limit of \( \displaystyle \frac{3 x + 1}{\sqrt{9 x^{2} + 4}} \) as \( x \to \infty \)

Problem 1.177 · medium

1. \[ \lim_{x \to \infty}\left(\frac{3 x + 1}{\sqrt{9 x^{2} + 4}}\right) \]
   limitStart with the original limit expression.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 x^{2} + 4}} + \frac{1}{\sqrt{9 x^{2} + 4}}\right) \]
   algebraSplit the fraction into two parts.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 x^{2} + 4}}\right) + \lim_{x \to \infty} \frac{1}{\sqrt{9 x^{2} + 4}} \]
   limit-lawApply the sum rule for limits.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 x^{2} + 4}}\right) \]
   limitThe second term approaches zero as x goes to infinity.✓ Proved
5. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 + \frac{4}{x^{2}}} \left|{x}\right|}\right) \]
   algebraFactor x**2 out of the square root.✓ Proved
6. \[ = \lim_{x \to \infty}\left(\frac{3}{\sqrt{9 + \frac{4}{x^{2}}}}\right) \]
   simplify algebraSimplify the expression inside the limit. Cancel the common factor of x.✓ Proved
7. \[ = 1 \]
   limit simplify simplifyEvaluate the limit as x approaches infinity. Simplify the resulting numerical expression. Final result.✓ Proved