Math Snap

STEP 1

Assumptions1. We are asked to find the limit as $x$ approaches3 of the function $\frac{x^{}-x-6}{x-3}$.
. We are dealing with real numbers.

STEP 2

First, we try to substitute $x =$ into the function to see if we can directly compute the limit.
$$\lim{x \rightarrow} \frac{x^{2}-x-6}{x-} = \frac{^{2}--6}{-}$$

STEP 3

Calculate the value after substitution.
$$\frac{3^{2}-3-6}{3-3} = \frac{9-3-6}{0}$$

STEP 4

implify the numerator.
$$\frac{9-3-6}{0} = \frac{0}{0}$$

STEP 5

We see that direct substitution leads to an indeterminate form ($\frac{0}{0}$). This means we need to do some algebraic manipulation to the function to find the limit. In this case, we can factor the numerator.

STEP 6

Factor the numerator $x^{2}-x-6$.
$$x^{2}-x-6 = (x-3)(x+2)$$

STEP 7

Substitute the factored form of the numerator back into the function.
$$\lim{x \rightarrow3} \frac{(x-3)(x+2)}{x-3}$$

STEP 8

We see that the term $(x-3)$ appears in both the numerator and the denominator, so we can cancel it out.
$$\lim{x \rightarrow3} x+2$$

STEP 9

Now that we have simplified the function, we can substitute $x =3$ into the function to find the limit.
$$\lim{x \rightarrow3} x+2 =3+2$$

Calculate the value after substitution.
$$3+2 =5$$So, the limit of the function as $x$ approaches3 is5.