QuestionFind the limits: 1) $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}-3 x}$ 2) $\lim _{x \rightarrow 3} \frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit of two different functions as x approaches a certain value. . We are dealing with real numbers.

STEP 2

Let's start with the first function. We need to find the limit as x approaches for the function $\frac{x^{2}-9}{x^{2}- x}$ .

STEP 3

First, we simplify the function. Notice that the numerator can be factored into $(x-3)(x+3)$ and the denominator can be factored into $x(x-3)$ .
$\frac{x^{2}-9}{x^{2}-3 x} = \frac{(x-3)(x+3)}{x(x-3)}$

STEP 4

Now, we can cancel out the common factor of $(x-3)$ from the numerator and the denominator.
$\frac{(x-3)(x+3)}{x(x-3)} = \frac{x+3}{x}$

STEP 5

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

STEP 6

Calculate the limit.
$\lim{x \rightarrow3} \frac{x+3}{x} = \frac{3+3}{3} =2$

STEP 7

Now, let's move on to the second function. We need to find the limit as x approaches0 for the function $\frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}$ .

STEP 8

First, we notice that if we substitute $x =0$ directly into the function, we get an indeterminate form of $\frac{0}{0}$ . This means we need to simplify the function first.

STEP 9

We can factor out a $3$ from the numerator and an $x^{3}$ from the denominator.
$\frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}} = \frac{3(x^{4}-2 x+4)}{x^{3}(x^{2}+4)}$

STEP 10

Now, we substitute $x =0$ into the simplified function to find the limit.
$\lim{x \rightarrow0} \frac{3(x^{4}-2 x+4)}{x^{3}(x^{2}+4)} = \frac{3(0^{4}-2 \cdot0+4)}{0^{3}(0^{2}+4)}$

STEP 11

Calculate the limit.
$\lim{x \rightarrow0} \frac{3(x^{4}- x+4)}{x^{3}(x^{}+4)} = \frac{3(0^{4}- \cdot0+4)}{0^{3}(0^{}+4)} = \frac{}{0}$

STEP 12

The limit as x approaches0 for the function $\frac{ x^{4}-6 x+12}{x^{5}+4 x^{}}$ is undefined because we are dividing by zero.
So, the solutions are. The limit as x approaches for the function $\frac{x^{2}-9}{x^{2}- x}$ is2.
2. The limit as x approaches0 for the function $\frac{ x^{4}-6 x+12}{x^{5}+4 x^{}}$ is undefined.

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