Math / Calculus

Question1. $\lim _{x \rightarrow \infty}\left[\left(x^{3}+x^{2}\right)^{\frac{1}{3}}-x\right]$

Studdy Solution

STEP 1

1. We need to find the limit of the given expression as $x$ approaches infinity.
2. The expression involves a cube root function and polynomial terms.
3. Simplifying the expression inside the limit will likely help in finding the limit as $x$ grows very large.

STEP 2

1. Rewrite the expression inside the limit to make the dominant term more apparent.
2. Simplify the expression by factoring out the dominant term.
3. Evaluate the limit of the simplified expression as $x$ approaches infinity.

STEP 3

Rewrite the given expression inside the limit to make the dominant term more apparent:
$\lim_{x \to \infty} \left[ \left(x^3 + x^2\right)^{\frac{1}{3}} - x \right]$
Factor out $x^3$ from inside the cube root:
$\lim_{x \to \infty} \left[ \left( x^3 \left( 1 + \frac{1}{x} \right) \right)^{\frac{1}{3}} - x \right]$

STEP 4

Simplify the expression inside the cube root:
$\left( x^3 \left( 1 + \frac{1}{x} \right) \right)^{\frac{1}{3}} = x \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}}$
So, the limit becomes:
$\lim_{x \to \infty} \left[ x \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} - x \right]$

STEP 5

Factor out $x$ from the expression inside the limit:
$\lim_{x \to \infty} x \left[ \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} - 1 \right]$

STEP 6

Use the first-order Taylor expansion for $\left( 1 + \frac{1}{x} \right)^{\frac{1}{3}}$ around $\frac{1}{x} = 0$ :
$\left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} \approx 1 + \frac{1}{3x}$
Therefore, the expression inside the limit becomes:
$\lim_{x \to \infty} x \left[ \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} - 1 \right] \approx \lim_{x \to \infty} x \left( \frac{1}{3x} \right)$

STEP 7

Simplify the expression further:
$\lim_{x \to \infty} \left( \frac{x}{3x} \right) = \lim_{x \to \infty} \left( \frac{1}{3} \right) = \frac{1}{3}$
Therefore, the limit is:
$\frac{1}{3}$

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Question1. $\lim _{x \rightarrow \infty}\left[\left(x^{3}+x^{2}\right)^{\frac{1}{3}}-x\right]$

Studdy Solution

STEP 1

1. We need to find the limit of the given expression as $x$ approaches infinity.
2. The expression involves a cube root function and polynomial terms.
3. Simplifying the expression inside the limit will likely help in finding the limit as $x$ grows very large.

STEP 2

1. Rewrite the expression inside the limit to make the dominant term more apparent.
2. Simplify the expression by factoring out the dominant term.
3. Evaluate the limit of the simplified expression as $x$ approaches infinity.

STEP 3

STEP 4

Simplify the expression inside the cube root:
$\left( x^3 \left( 1 + \frac{1}{x} \right) \right)^{\frac{1}{3}} = x \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}}$
So, the limit becomes:
$\lim_{x \to \infty} \left[ x \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} - x \right]$

STEP 5

Factor out $x$ from the expression inside the limit:
$\lim_{x \to \infty} x \left[ \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} - 1 \right]$

STEP 6

STEP 7

Simplify the expression further:
$\lim_{x \to \infty} \left( \frac{x}{3x} \right) = \lim_{x \to \infty} \left( \frac{1}{3} \right) = \frac{1}{3}$
Therefore, the limit is:
$\frac{1}{3}$

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Question1. lim⁡x→∞[(x3+x2)13−x]\lim _{x \rightarrow \infty}\left[\left(x^{3}+x^{2}\right)^{\frac{1}{3}}-x\right]limx→∞​[(x3+x2)31​−x]

Studdy Solution

STEP 1

1. We need to find the limit of the given expression as x x x approaches infinity.2. The expression involves a cube root function and polynomial terms.3. Simplifying the expression inside the limit will likely help in finding the limit as x x x grows very large.

STEP 2

1. Rewrite the expression inside the limit to make the dominant term more apparent.2. Simplify the expression by factoring out the dominant term.3. Evaluate the limit of the simplified expression as x x x approaches infinity.

STEP 3

STEP 4

STEP 5

Factor out x x x from the expression inside the limit:lim⁡x→∞x[(1+1x)13−1] \lim_{x \to \infty} x \left[ \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} - 1 \right] x→∞lim​x[(1+x1​)31​−1]

STEP 6

STEP 7

Simplify the expression further:lim⁡x→∞(x3x)=lim⁡x→∞(13)=13 \lim_{x \to \infty} \left( \frac{x}{3x} \right) = \lim_{x \to \infty} \left( \frac{1}{3} \right) = \frac{1}{3} x→∞lim​(3xx​)=x→∞lim​(31​)=31​Therefore, the limit is:13 \frac{1}{3} 31​

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Question1. $\lim _{x \rightarrow \infty}\left[\left(x^{3}+x^{2}\right)^{\frac{1}{3}}-x\right]$

1. We need to find the limit of the given expression as $x$ approaches infinity.
2. The expression involves a cube root function and polynomial terms.
3. Simplifying the expression inside the limit will likely help in finding the limit as $x$ grows very large.

1. Rewrite the expression inside the limit to make the dominant term more apparent.
2. Simplify the expression by factoring out the dominant term.
3. Evaluate the limit of the simplified expression as $x$ approaches infinity.

Factor out $x$ from the expression inside the limit:
$\lim_{x \to \infty} x \left[ \left( 1 + \frac{1}{x} \right)^{\frac{1}{3}} - 1 \right]$

Simplify the expression further:
$\lim_{x \to \infty} \left( \frac{x}{3x} \right) = \lim_{x \to \infty} \left( \frac{1}{3} \right) = \frac{1}{3}$
Therefore, the limit is:
$\frac{1}{3}$