Math / Algebra

QuestionGive the equation of the oblique asymptote, if any, of the function. $f(x)=\frac{x^{2}-7}{49 x-x^{4}}$

Studdy Solution

STEP 1

What is this asking? We need to find the equation of the oblique asymptote, which is a slanted straight line that the function gets really close to as $x$ becomes super large or super small. Watch out! Oblique asymptotes only happen when the degree of the numerator is exactly one more than the degree of the denominator.
Also, remember that an asymptote isn't part of the function itself, it's just a line that the function approaches.

STEP 2

1. Check the degrees
2. Polynomial long division
3. Identify the oblique asymptote

STEP 3

Let's look at our function $f(x) = \frac{x^2 - 7}{49x - x^4}$ .
The highest power of $x$ in the numerator, $x^2 - 7$ , is $2$ .
We call this the degree of the numerator.

STEP 4

The highest power of $x$ in the denominator, $49x - x^4$ , is $4$ .
We call this the degree of the denominator.

STEP 5

For an oblique asymptote to exist, the degree of the numerator needs to be exactly one greater than the degree of the denominator.
Here, the degree of the numerator is $\bf{2}$ and the degree of the denominator is $\bf{4}$ .
Since $2$ is not one greater than $4$ , there is no oblique asymptote!

STEP 6

Even though we know there's no oblique asymptote, let's quickly see what would happen if we did polynomial long division.
We would rewrite our function to prepare for division: $f(x) = \frac{x^2 - 7}{-x^4 + 49x + 0x^3 + 0x^2}$ .
Notice how we added $+ 0x^3 + 0x^2$ .
Adding zero doesn't change the value, but it helps keep things organized during the division.

STEP 7

If we were to perform the division, the result would be $0$ with a remainder of $x^2 - 7$ .
Since the quotient is $0$ , this further confirms that there's no oblique asymptote.

STEP 8

As we saw in the previous steps, the degrees of the numerator and denominator tell us there's no oblique asymptote, and the polynomial long division confirms this.

STEP 9

There is no oblique asymptote for the function $f(x) = \frac{x^2 - 7}{49x - x^4}$ .

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QuestionGive the equation of the oblique asymptote, if any, of the function. $f(x)=\frac{x^{2}-7}{49 x-x^{4}}$

Studdy Solution

STEP 1

STEP 2

1. Check the degrees
2. Polynomial long division
3. Identify the oblique asymptote

STEP 3

Let's look at our function $f(x) = \frac{x^2 - 7}{49x - x^4}$ .
The highest power of $x$ in the numerator, $x^2 - 7$ , is $2$ .
We call this the degree of the numerator.

STEP 4

The highest power of $x$ in the denominator, $49x - x^4$ , is $4$ .
We call this the degree of the denominator.

STEP 5

For an oblique asymptote to exist, the degree of the numerator needs to be exactly one greater than the degree of the denominator.
Here, the degree of the numerator is $\bf{2}$ and the degree of the denominator is $\bf{4}$ .
Since $2$ is not one greater than $4$ , there is no oblique asymptote!

STEP 6

STEP 7

If we were to perform the division, the result would be $0$ with a remainder of $x^2 - 7$ .
Since the quotient is $0$ , this further confirms that there's no oblique asymptote.

STEP 8

As we saw in the previous steps, the degrees of the numerator and denominator tell us there's no oblique asymptote, and the polynomial long division confirms this.

STEP 9

There is no oblique asymptote for the function $f(x) = \frac{x^2 - 7}{49x - x^4}$ .

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QuestionGive the equation of the oblique asymptote, if any, of the function. f(x)=x2−749x−x4f(x)=\frac{x^{2}-7}{49 x-x^{4}}f(x)=49x−x4x2−7​

Studdy Solution

STEP 1

STEP 2

1. Check the degrees2. Polynomial long division3. Identify the oblique asymptote

STEP 3

Let's look at our function f(x)=x2−749x−x4f(x) = \frac{x^2 - 7}{49x - x^4}f(x)=49x−x4x2−7​.The **highest power of** xxx in the numerator, x2−7x^2 - 7x2−7, is 222.We call this the **degree of the numerator**.

STEP 4

The **highest power of** xxx in the denominator, 49x−x449x - x^449x−x4, is 444.We call this the **degree of the denominator**.

STEP 5

For an oblique asymptote to exist, the degree of the numerator needs to be *exactly* one greater than the degree of the denominator.Here, the degree of the numerator is 2\bf{2}2 and the degree of the denominator is 4\bf{4}4.Since 222 is *not* one greater than 444, there is *no* oblique asymptote!

STEP 6

STEP 7

If we were to perform the division, the result would be 000 with a remainder of x2−7x^2 - 7x2−7.Since the quotient is 000, this further confirms that there's no oblique asymptote.

STEP 8

As we saw in the previous steps, the degrees of the numerator and denominator tell us there's no oblique asymptote, and the polynomial long division confirms this.

STEP 9

There is **no oblique asymptote** for the function f(x)=x2−749x−x4f(x) = \frac{x^2 - 7}{49x - x^4}f(x)=49x−x4x2−7​.

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QuestionGive the equation of the oblique asymptote, if any, of the function. $f(x)=\frac{x^{2}-7}{49 x-x^{4}}$

1. Check the degrees
2. Polynomial long division
3. Identify the oblique asymptote

Let's look at our function $f(x) = \frac{x^2 - 7}{49x - x^4}$ .
The highest power of $x$ in the numerator, $x^2 - 7$ , is $2$ .
We call this the degree of the numerator.

The highest power of $x$ in the denominator, $49x - x^4$ , is $4$ .
We call this the degree of the denominator.

For an oblique asymptote to exist, the degree of the numerator needs to be exactly one greater than the degree of the denominator.
Here, the degree of the numerator is $\bf{2}$ and the degree of the denominator is $\bf{4}$ .
Since $2$ is not one greater than $4$ , there is no oblique asymptote!

If we were to perform the division, the result would be $0$ with a remainder of $x^2 - 7$ .
Since the quotient is $0$ , this further confirms that there's no oblique asymptote.

There is no oblique asymptote for the function $f(x) = \frac{x^2 - 7}{49x - x^4}$ .