(1) Evaluate the following limits. State the asymptotes, if any: (i) \( \lim _{t \rightarrow 3} \frac{2-t}{-2 t^{2}+7 t-3} \) (ii) \( \lim _{x \rightarrow-\infty} \frac{x-\sqrt{x^{2}-x+1}}{2 x-\sqrt{4 x^{2}+5 x-1}} \)

Let's evaluate each limit and determine any asymptotes associated with them. --- **(i)** \[ \lim_{t \rightarrow 3} \frac{2 - t}{-2t^{2} + 7t - 3} \] **Step 1: Factor the Denominator** First, factor the quadratic in the denominator: \[ -2t^{2} + 7t - 3 = -(2t^{2} - 7t + 3) \] Factor \(2t^{2} - 7t + 3\): \[ 2t^{2} - 7t + 3 = (2t - 1)(t - 3) \] So, the denominator becomes: \[ -2t^{2} + 7t - 3 = -(2t - 1)(t - 3) \] **Step 2: Analyze the Limit** As \( t \rightarrow 3 \): - The **numerator**: \(2 - t \rightarrow -1\). - The **denominator**: \(-(2t - 1)(t - 3) \rightarrow -[(6 - 1)(0)] = 0\). Thus, the expression approaches \(\frac{-1}{0}\), suggesting a vertical asymptote at \( t = 3 \). **Step 3: Determine the Direction of the Asymptote** - **From the Right (\( t \rightarrow 3^+ \))**: - \( t - 3 \) is positive. - Denominator approaches \(0^+\). - The limit approaches \( \frac{-1}{0^+} = -\infty \). - **From the Left (\( t \rightarrow 3^- \))**: - \( t - 3 \) is negative. - Denominator approaches \(0^-\). - The limit approaches \( \frac{-1}{0^-} = +\infty \). **Conclusion for (i):** \[ \lim_{t \rightarrow 3} \frac{2 - t}{-2t^{2} + 7t - 3} \text{ does not exist because it approaches } +\infty \text{ from the left and } -\infty \text{ from the right.} \] **Asymptote:** There is a **vertical asymptote** at \( t = 3 \). --- **(ii)** \[ \lim_{x \rightarrow -\infty} \frac{x - \sqrt{x^{2} - x + 1}}{2x - \sqrt{4x^{2} + 5x - 1}} \] **Step 1: Simplify the Expression** For large \(|x|\), especially as \( x \rightarrow -\infty \): - **Numerator:** \[ x - \sqrt{x^{2} - x + 1} \approx x - (-x) \sqrt{1 - \frac{1}{x} + \frac{1}{x^{2}}} = x + x\left(1 - \frac{1}{2x} + \cdots\right) \approx 2x - \frac{1}{2} \] - **Denominator:** \[ 2x - \sqrt{4x^{2} + 5x - 1} \approx 2x - (-2x) \sqrt{1 + \frac{5}{4x} - \frac{1}{4x^{2}}} = 2x + 2x\left(1 + \frac{5}{8x} + \cdots\right) \approx 4x + \frac{5}{4} \] **Step 2: Compute the Limit** \[ \lim_{x \rightarrow -\infty} \frac{2x - \frac{1}{2}}{4x + \frac{5}{4}} = \lim_{x \rightarrow -\infty} \frac{2x}{4x} = \frac{1}{2} \] **Conclusion for (ii):** \[ \lim_{x \rightarrow -\infty} \frac{x - \sqrt{x^{2} - x + 1}}{2x - \sqrt{4x^{2} + 5x - 1}} = \frac{1}{2} \] **Asymptote:** There is a **horizontal asymptote** at \( y = \frac{1}{2} \). --- **Final Summary:** 1. **Limit (i):** \[ \lim_{t \rightarrow 3} \frac{2 - t}{-2t^{2} + 7t - 3} \text{ does not exist } (\text{approaches } +\infty \text{ from the left and } -\infty \text{ from the right}). \] **Asymptote:** Vertical asymptote at \( t = 3 \). 2. **Limit (ii):** \[ \lim_{x \rightarrow -\infty} \frac{x - \sqrt{x^{2} - x + 1}}{2x - \sqrt{4x^{2} + 5x - 1}} = \frac{1}{2}. \] **Asymptote:** Horizontal asymptote at \( y = \frac{1}{2} \).