Answer
As \( t \) approaches 3, the limit of \( \frac{2 - t}{-2t^2 + 7t - 3} \) does not exist because it approaches \( +\infty \) from the right and \( -\infty \) from the left. The function has vertical asymptotes at \( t = \frac{1}{2} \) and \( t = 3 \), and a horizontal asymptote at \( y = 0 \).
Solution
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 \( y = 0 \).
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
