3) $$ \lim _{x \rightarrow-5} \frac{x^{2}-25}{(x+5)(x-6)} $$

Question

UpStudy AI · Accepted Answer

To evaluate the limit:

$$
\lim_{x \rightarrow -5} \frac{x^{2} - 25}{(x + 5)(x - 6)}
$$

**Step 1: Factor the numerator.**

Notice that $$x^2 - 25$$ is a difference of squares, which can be factored as:

$$
x^2 - 25 = (x - 5)(x + 5)
$$

**Step 2: Simplify the expression.**

Substitute the factored form back into the original limit:

$$
\frac{(x - 5)(x + 5)}{(x + 5)(x - 6)} = \frac{x - 5}{x - 6} \quad \text{(for } x \neq -5 \text{)}
$$

**Step 3: Evaluate the limit.**

Now, compute the limit by substituting $$x = -5$$ into the simplified expression:

$$
\frac{-5 - 5}{-5 - 6} = \frac{-10}{-11} = \frac{10}{11}
$$

**Final Answer:**

$$
\boxed{\dfrac{10}{11}}
$$
The limit is $$ \frac{10}{11} $$.

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