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

To evaluate the limit and identify the asymptotes for the function: \[ \lim_{t \to 3} \frac{2 - t}{-2t^2 + 7t - 3} \] ### Step 1: Simplify the Expression First, factor the denominator: \[ -2t^2 + 7t - 3 = - (2t^2 - 7t + 3) \] Factor the quadratic \(2t^2 - 7t + 3\): \[ 2t^2 - 7t + 3 = (2t - 1)(t - 3) \] So, the original expression becomes: \[ \frac{2 - t}{-2t^2 + 7t - 3} = \frac{-(t - 2)}{- (2t - 1)(t - 3)} = \frac{t - 2}{(2t - 1)(t - 3)} \] ### Step 2: Evaluate the Limit As \(t\) approaches 3: - **Numerator:** \(t - 2 \to 3 - 2 = 1\) - **Denominator:** \((2t - 1)(t - 3) \to (6 - 1)(0) = 5 \times 0 = 0\) Since the denominator approaches 0 and the numerator approaches 1, the behavior of the limit depends on the direction from which \(t\) approaches 3. - **From the Right (\(t \to 3^+\)):** - \(t - 3 \) is positive. - The denominator approaches \(0^+\), making the entire fraction tend to \(+\infty\). - **From the Left (\(t \to 3^-\)):** - \(t - 3 \) is negative. - The denominator approaches \(0^-\), making the entire fraction tend to \(-\infty\). Since the left-hand limit and right-hand limit are not equal (they approach \(+\infty\) and \(-\infty\) respectively), the overall limit **does not exist**. ### Step 3: Identify Asymptotes 1. **Vertical Asymptotes:** - Occur where the denominator is zero and the numerator is not zero. - From the factored form, \( (2t - 1)(t - 3) = 0 \): - \(2t - 1 = 0 \Rightarrow t = \frac{1}{2}\) - \(t - 3 = 0 \Rightarrow t = 3\) - **Vertical Asymptotes at:** \( t = \frac{1}{2} \) and \( t = 3 \) 2. **Horizontal Asymptote:** - Compare the degrees of the numerator and the denominator. - The numerator is degree 1, and the denominator is degree 2. - As \( t \to \pm\infty \), the function approaches \(0\). - **Horizontal Asymptote at:** \( y = 0 \) ### Summary - **Limit Evaluation:** \[ \lim_{t \to 3} \frac{2 - t}{-2t^2 + 7t - 3} \text{ does not exist because it approaches } +\infty \text{ from the right and } -\infty \text{ from the left.} \] - **Asymptotes:** - **Vertical Asymptotes:** \( t = \frac{1}{2} \) and \( t = 3 \) - **Horizontal Asymptote:** \( y = 0 \) **Final Answer:** (i) As \( t \) approaches 3, (2 – t)/(–2t² + 7t – 3) grows without bound: the limit is +∞ from above and –∞ from below. The graph has vertical asymptotes at \( t = \tfrac{1}{2} \) and \( t = 3 \), and a horizontal asymptote at \( y = 0 \).