Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)
 
lim 
t→0 

5
t
 − 
5
t2 + t

 

We are asked to evaluate the following limit:

\[
\lim_{{t \to 0}} \left( \frac{5}{t} - \frac{5}{t^2 + t} \right)
\]

### Step-by-step solution:

1. **Rewrite the expression:**
   The limit can be rewritten as:

   \[
   \lim_{{t \to 0}} \left( \frac{5}{t} - \frac{5}{t^2 + t} \right) = 5 \cdot \lim_{{t \to 0}} \left( \frac{1}{t} - \frac{1}{t^2 + t} \right)
   \]

2. **Find a common denominator:**
   To simplify the expression inside the limit, find a common denominator:

   \[
   \frac{1}{t} - \frac{1}{t^2 + t} = \frac{(t^2 + t) - t}{t(t^2 + t)}
   \]

   Simplifying the numerator:

   \[
   (t^2 + t) - t = t^2
   \]

   So, the expression becomes:

   \[
   \frac{t^2}{t(t^2 + t)} = \frac{t}{t^2 + t}
   \]

3. **Simplify the limit:**
   Now, the original limit becomes:

   \[
   5 \cdot \lim_{{t \to 0}} \frac{t}{t^2 + t}
   \]

   Factor the denominator:

   \[
   \lim_{{t \to 0}} \frac{t}{t(t + 1)} = \lim_{{t \to 0}} \frac{1}{t + 1}
   \]

4. **Evaluate the limit:**
   As \( t \to 0 \), \( \frac{1}{t + 1} \) approaches \( \frac{1}{1} = 1 \).

   Therefore, the limit is:

   \[
   5 \times 1 = 5
   \]

### Final answer:
\[
\lim_{{t \to 0}} \left( \frac{5}{t} - \frac{5}{t^2 + t} \right) = 5
\]

Would you like more details on any of the steps?

Here are 5 related questions to further your understanding:
1. How do you find common denominators for more complex fractions?
2. What are the general steps for evaluating limits as \( t \to 0 \)?
3. Can you explain how to factor quadratic expressions in more detail?
4. How do limits involving infinity differ from those involving zero?
5. How does L'Hopital's rule apply to indeterminate forms like \(\frac{0}{0}\)?

**Tip:** Always check if you can simplify the expression before applying limit rules.

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim t→0 (5/t) − (5/(t^2 + t))

This detailed solution evaluates the limit of (5/t) - (5/(t^2 + t)) as t approaches 0, covering key concepts like common denominators, algebraic simplification, and limit laws. Suitable for high school students in Grades 11-12 studying Calculus.

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