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Math  /  Calculus

Questiond) lim⁡x→−∞3x3−2x2+1x2+1\lim _{x \rightarrow-\infty} \frac{3 x^{3}-2 x^{2}+1}{x^{2}+1}

Studdy Solution

STEP 1

What is this asking? We need to find the limit of a rational function as xx approaches negative infinity. Watch out! Don't forget to consider the degrees of the polynomials in the numerator and denominator!

STEP 2

1. Analyze the degrees
2. Divide by the highest power of xx
3. Evaluate the limit

STEP 3

Alright, let's **look** at the degrees of the polynomials!
We've got a **cubic** polynomial 3x3−2x2+13x^3 - 2x^2 + 1 in the numerator and a **quadratic** polynomial x2+1x^2 + 1 in the denominator.
Since the degree of the numerator is **greater** than the degree of the denominator, we expect the limit as xx approaches negative infinity to be either positive or negative infinity.

STEP 4

The **highest power** of xx in the entire expression is x3x^3.
Let's **divide** both the numerator and the denominator by x3x^3 to simplify things!
This is a **clever trick** because it helps us see what happens when xx gets **really big** (or really small, in this case!).

STEP 5

lim⁡x→−∞3x3−2x2+1x2+1=lim⁡x→−∞3x3x3−2x2x3+1x3x2x3+1x3 \lim_{x \to -\infty} \frac{3x^3 - 2x^2 + 1}{x^2 + 1} = \lim_{x \to -\infty} \frac{\frac{3x^3}{x^3} - \frac{2x^2}{x^3} + \frac{1}{x^3}}{\frac{x^2}{x^3} + \frac{1}{x^3}}

STEP 6

Now, let's **simplify** that expression! lim⁡x→−∞3−2x+1x31x+1x3 \lim_{x \to -\infty} \frac{3 - \frac{2}{x} + \frac{1}{x^3}}{\frac{1}{x} + \frac{1}{x^3}}

STEP 7

As xx approaches negative infinity, terms like 2x\frac{2}{x}, 1x3\frac{1}{x^3}, and 1x\frac{1}{x} all approach **zero**.
Think about it: as the denominator gets **super huge** (or super small, since it's negative), the whole fraction gets **tiny**!

STEP 8

So, we can **rewrite** our limit as: lim⁡x→−∞3−2x+1x31x+1x3=3−0+00+0=30 \lim_{x \to -\infty} \frac{3 - \frac{2}{x} + \frac{1}{x^3}}{\frac{1}{x} + \frac{1}{x^3}} = \frac{3 - 0 + 0}{0 + 0} = \frac{3}{0}

STEP 9

Now, we've got a **3** in the numerator and a **0** in the denominator.
This means our limit is **unbounded**!
But we need to figure out if it's going to positive or negative infinity.

STEP 10

Since xx is approaching *negative* infinity, the terms 1x\frac{1}{x} and 1x3\frac{1}{x^3} will be **small negative numbers**.
So, the denominator is approaching zero from the **negative side**.
Since the numerator is positive (**3**) and the denominator is approaching zero from the negative side, the whole fraction is going towards **negative infinity**!

STEP 11

The limit of the function as xx approaches negative infinity is −∞-\infty.

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