Math / Calculus

QuestionQ35) $\lim _{x \rightarrow \infty} \frac{(3 x-1)^{2}\left(2 x^{2}+1\right)}{5 x(3 x+3)^{3}}$ ? a) $\frac{15}{5}$ b) $\frac{2}{5}$ c) $\frac{2}{15}$ d) None

Studdy Solution

STEP 1

1. We are dealing with a limit as $x$ approaches infinity.
2. The expression involves polynomials, so we will use the method of dominant terms to simplify the limit.

STEP 2

1. Identify the dominant terms in the numerator and the denominator.
2. Simplify the expression using the dominant terms.
3. Calculate the limit of the simplified expression as $x$ approaches infinity.

STEP 3

Identify the dominant terms in the numerator and the denominator. The dominant term in a polynomial is the term with the highest power of $x$ .
- In the numerator $(3x - 1)^2(2x^2 + 1)$ , the dominant term is $(3x)^2 \cdot 2x^2 = 18x^4$ . - In the denominator $5x(3x + 3)^3$ , the dominant term is $5x \cdot (3x)^3 = 135x^4$ .

STEP 4

Simplify the expression by using only the dominant terms:
$\frac{(3x - 1)^2(2x^2 + 1)}{5x(3x + 3)^3} \approx \frac{18x^4}{135x^4}$

STEP 5

Calculate the limit of the simplified expression as $x$ approaches infinity:
$\lim_{x \to \infty} \frac{18x^4}{135x^4} = \frac{18}{135} = \frac{2}{15}$
The value of the limit is:
$\boxed{\frac{2}{15}}$

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