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 the numerator $$ x^2 - 25 $$ is a difference of squares and can be factored as:

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

**Step 2: Simplify the Expression**

Substitute the factored form back into the limit:

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

Cancel the common factor $$ (x + 5) $$ in the numerator and denominator (provided $$ x \neq -5 $$, which is valid since we're taking a limit):

$$
\lim_{x \rightarrow -5} \frac{x - 5}{x - 6}
$$

**Step 3: Substitute $$ x = -5 $$**

Now, substitute $$ x = -5 $$ into the simplified expression:

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

**Final Answer:**

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

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