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Math  /  Calculus

QuestionFind the limit, if it exists. (If an answer does not exist, enter DNE.) lim⁡x→∞x+3x22x−1\lim _{x \rightarrow \infty} \frac{\sqrt{x+3 x^{2}}}{2 x-1}

Studdy Solution

STEP 1

1. The limit involves a rational function with a square root in the numerator and a polynomial in the denominator.
2. The behavior of the function as x x approaches infinity needs to be analyzed.
3. Simplification of the expression under the limit is necessary to identify leading terms.
4. The limit can be evaluated by comparing the degrees of the leading terms in the numerator and the denominator.

STEP 2

1. Simplify the expression under the square root in the numerator.
2. Identify the leading terms in both the numerator and the denominator.
3. Simplify the fraction by dividing the numerator and the denominator by the highest power of x x present in the denominator.
4. Evaluate the limit of the simplified expression as x x approaches infinity.

STEP 3

Simplify the expression under the square root in the numerator.
x+3x2 \sqrt{x + 3x^2}

STEP 4

Factor out x2 x^2 from the expression inside the square root to simplify it.
x+3x2=x(1+3x)=x⋅1+3x \sqrt{x + 3x^2} = \sqrt{x(1 + 3x)} = \sqrt{x} \cdot \sqrt{1 + 3x}

STEP 5

Simplify the square root expression further.
For large x x : x⋅1+3x=x⋅3x(13x+1)=x⋅3x⋅13x+1 \sqrt{x} \cdot \sqrt{1 + 3x} = \sqrt{x} \cdot \sqrt{3x\left(\frac{1}{3x} + 1\right)} = \sqrt{x} \cdot \sqrt{3x} \cdot \sqrt{\frac{1}{3x} + 1}

STEP 6

Combine the square roots and simplify:
x⋅3x=3x2=3⋅x \sqrt{x} \cdot \sqrt{3x} = \sqrt{3x^2} = \sqrt{3} \cdot x

STEP 7

Identify the leading terms in both the numerator and the denominator.
Numerator: 3⋅x \sqrt{3} \cdot x
Denominator: 2x−1 2x - 1

STEP 8

Divide both the numerator and the denominator by the highest power of x x in the denominator, which is x x .
3⋅x2x−1=3⋅x/x(2x−1)/x=32−1x \frac{\sqrt{3} \cdot x}{2x - 1} = \frac{\sqrt{3} \cdot x / x}{(2x - 1) / x} = \frac{\sqrt{3}}{2 - \frac{1}{x}}

STEP 9

Evaluate the limit of the simplified expression as x x approaches infinity.
lim⁡x→∞32−1x \lim_{x \to \infty} \frac{\sqrt{3}}{2 - \frac{1}{x}}

STEP 10

As x x approaches infinity, 1x \frac{1}{x} approaches 0.
lim⁡x→∞32−0=32 \lim_{x \to \infty} \frac{\sqrt{3}}{2 - 0} = \frac{\sqrt{3}}{2}
Solution: The limit is:
lim⁡x→∞x+3x22x−1=32 \lim_{x \to \infty} \frac{\sqrt{x + 3x^2}}{2x - 1} = \frac{\sqrt{3}}{2}

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