QuestionFind the limit as $x$ approaches 3 for the expression $\frac{x^{2}-9}{x^{2}-3x}$ .

Studdy Solution

STEP 1

Assumptions1. We are trying to find the limit as x approaches3 of the function $\frac{x^{}-9}{x^{}-3 x}$ . We will use algebraic simplification and limit properties to find the limit.

STEP 2

First, let's try to simplify the function. We can do this by factoring the numerator and the denominator.
The numerator $x^{2}-9$ can be factored into $(x-)(x+)$ using the difference of squares formula.
The denominator $x^{2}-x$ can be factored into $x(x-)$ by taking out the common factor of x.

STEP 3

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

STEP 4

We can see that the factor $(x-3)$ is common to both the numerator and the denominator. We can cancel out this common factor.
$\lim{x \rightarrow3} \frac{x+3}{x}$

STEP 5

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

STEP 6

Calculate the limit.
$\lim{x \rightarrow3} \frac{x+3}{x} = \frac{6}{3} =2$ The limit of the function as x approaches3 is2.

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