Question

If $f(x) = \frac{1 - x + \sqrt{9x^2 + 10x + 1}}{2x}$, then $\lim_{x \to -1^-} f(x) =$

• A. $1$
• B. $-1$
• C. $0$
• D. $-1/5$

Hint:

Limits Involving Square Roots:

• Always analyze root domain and behavior near point.
• Use factorization to simplify.
• Watch sign direction in one-sided limits.

Answer 1

The Correct Option is B

Let $P(x) = 9x^2 + 10x + 1 = (x+1)(9x+1)$ As $x \to -1^-$: $(x+1) \to 0^-$ and $(9x+1) \to -8$ $\Rightarrow$ root term is real and positive. \[ f(x) \to \frac{1 - (-1) + 0}{2(-1)} = \frac{2}{-2} = -1 \]