(1) Evaluate the following limits. State the asymptotes, if any:
(i)

Ask by Hampton Vaughn. in Canada

Jan 26,2025

Question

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

UpStudy AI · Accepted Answer

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 $$ y = 0 $$.
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 $$ y = 0 $$.

(1) Evaluate the following limits. State the asymptotes, if any:
(i)

Upstudy AI Solution

Answer

Solution

Step 1: Simplify the Expression

Step 2: Evaluate the Limit

Step 3: Identify Asymptotes

Summary

Extra Insights

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