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. \[ f(x) \to \frac{1 - (-1) + 0}{2(-1)} = \frac{2}{-2} = -1 \]

Answer 2

Step 1: Understand the function and limit to be found
We need to find the left-hand limit of
\( f(x) = \frac{1 - x + \sqrt{9x^2 + 10x + 1}}{2x} \) as \( x \to -1^- \).
This means we approach \( -1 \) from values less than \(-1\).

Step 2: Check the expression under the square root
Inside the square root: \(9x^2 + 10x + 1\).
At \(x = -1\), this equals \(9(1) + 10(-1) + 1 = 9 - 10 + 1 = 0\).
As \(x \to -1\), the expression under the root approaches zero, so the square root will approach 0.

Step 3: Simplify expression near \(x = -1\)
Substitute \( x = -1 + h \) where \( h \to 0^- \) (i.e., \(h\) is a small negative number).
Then:
\[ f(-1 + h) = \frac{1 - (-1 + h) + \sqrt{9(-1 + h)^2 + 10(-1 + h) + 1}}{2(-1 + h)} \] Simplify numerator first:
\[ 1 + 1 - h + \sqrt{9(1 - 2h + h^2) - 10 + 10h + 1} = 2 - h + \sqrt{9 - 18h + 9h^2 - 10 + 10h + 1} \] Inside the root:
\[ (9 - 10 + 1) + (-18h + 10h) + 9h^2 = 0 - 8h + 9h^2 = -8h + 9h^2 \]
For very small \(h\), \(9h^2\) is negligible compared to \(-8h\). Since \(h \to 0^-\), \(-8h\) is positive (because \(h\) is negative), so the root is real and approximately \(\sqrt{-8h}\).

Step 4: Approximate square root term
\[ \sqrt{-8h} = \sqrt{8|h|} \quad \text{since } h < 0 \] The numerator near \(x = -1\) becomes:
\[ 2 - h + \sqrt{8|h|} \] The denominator is:
\[ 2(-1 + h) = -2 + 2h \]

Step 5: Take the limit as \( h \to 0^- \)
As \( h \to 0^- \), \(-h \to 0^+\) and \(\sqrt{8|h|} \to 0\). So numerator approaches \(2 + 0 + 0 = 2\).
Denominator approaches \(-2 + 0 = -2\).

Step 6: Final limit value
\[ \lim_{x \to -1^-} f(x) = \frac{2}{-2} = -1 \]

Conclusion:
The left-hand limit of the function as \( x \to -1^- \) is \(-1\).