QuestionFind the limit as $x$ approaches infinity for $\frac{4x - 1}{2x + 1}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches infinity for the function $\frac{4x-1}{x+1}$ . . We are given that the limit equals.

STEP 2

We start by writing down the given limit expression.
$\lim{x \rightarrow+\infty} \frac{4 x-1}{2 x+1}$

STEP 3

We divide the numerator and the denominator by $x$ , which is the highest power of $x$ in the denominator.
$\lim{x \rightarrow+\infty} \frac{\frac{x}{x}-\frac{1}{x}}{\frac{2x}{x}+\frac{1}{x}}$

STEP 4

implify the expression.
$\lim{x \rightarrow+\infty} \frac{4-\frac{1}{x}}{2+\frac{1}{x}}$

STEP 5

As $x$ approaches infinity, $\frac{1}{x}$ approaches0. So, we can replace $\frac{1}{x}$ with0 in the expression.
$\lim{x \rightarrow+\infty} \frac{4-0}{2+0}$

STEP 6

implify the expression to find the limit.
$\lim{x \rightarrow+\infty} \frac{4}{2} =2$ The limit of the function as $x$ approaches infinity is indeed2.

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QuestionFind the limit as $x$ approaches infinity for $\frac{4x - 1}{2x + 1}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches infinity for the function $\frac{4x-1}{x+1}$ . . We are given that the limit equals.

STEP 2

We start by writing down the given limit expression.
$\lim{x \rightarrow+\infty} \frac{4 x-1}{2 x+1}$

STEP 3

We divide the numerator and the denominator by $x$ , which is the highest power of $x$ in the denominator.
$\lim{x \rightarrow+\infty} \frac{\frac{x}{x}-\frac{1}{x}}{\frac{2x}{x}+\frac{1}{x}}$

STEP 4

implify the expression.
$\lim{x \rightarrow+\infty} \frac{4-\frac{1}{x}}{2+\frac{1}{x}}$

STEP 5

As $x$ approaches infinity, $\frac{1}{x}$ approaches0. So, we can replace $\frac{1}{x}$ with0 in the expression.
$\lim{x \rightarrow+\infty} \frac{4-0}{2+0}$

STEP 6

implify the expression to find the limit.
$\lim{x \rightarrow+\infty} \frac{4}{2} =2$ The limit of the function as $x$ approaches infinity is indeed2.

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QuestionFind the limit as xxx approaches infinity for 4x−12x+1\frac{4x - 1}{2x + 1}2x+14x−1​.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches infinity for the function 4x−1x+1\frac{4x-1}{x+1}x+14x−1​. . We are given that the limit equals.

STEP 2

We start by writing down the given limit expression.lim⁡x→+∞4x−12x+1\lim{x \rightarrow+\infty} \frac{4 x-1}{2 x+1}limx→+∞2x+14x−1​

STEP 3

We divide the numerator and the denominator by xxx, which is the highest power of xxx in the denominator.lim⁡x→+∞xx−1x2xx+1x\lim{x \rightarrow+\infty} \frac{\frac{x}{x}-\frac{1}{x}}{\frac{2x}{x}+\frac{1}{x}}limx→+∞x2x​+x1​xx​−x1​​

STEP 4

implify the expression.lim⁡x→+∞4−1x2+1x\lim{x \rightarrow+\infty} \frac{4-\frac{1}{x}}{2+\frac{1}{x}}limx→+∞2+x1​4−x1​​

STEP 5

As xxx approaches infinity, 1x\frac{1}{x}x1​ approaches0. So, we can replace 1x\frac{1}{x}x1​ with0 in the expression.lim⁡x→+∞4−02+0\lim{x \rightarrow+\infty} \frac{4-0}{2+0}limx→+∞2+04−0​

STEP 6

implify the expression to find the limit.lim⁡x→+∞42=2\lim{x \rightarrow+\infty} \frac{4}{2} =2limx→+∞24​=2The limit of the function as xxx approaches infinity is indeed2.

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QuestionFind the limit as $x$ approaches infinity for $\frac{4x - 1}{2x + 1}$ .

Assumptions1. We are asked to find the limit as x approaches infinity for the function $\frac{4x-1}{x+1}$ . . We are given that the limit equals.

We start by writing down the given limit expression.
$\lim{x \rightarrow+\infty} \frac{4 x-1}{2 x+1}$

We divide the numerator and the denominator by $x$ , which is the highest power of $x$ in the denominator.
$\lim{x \rightarrow+\infty} \frac{\frac{x}{x}-\frac{1}{x}}{\frac{2x}{x}+\frac{1}{x}}$

implify the expression.
$\lim{x \rightarrow+\infty} \frac{4-\frac{1}{x}}{2+\frac{1}{x}}$

As $x$ approaches infinity, $\frac{1}{x}$ approaches0. So, we can replace $\frac{1}{x}$ with0 in the expression.
$\lim{x \rightarrow+\infty} \frac{4-0}{2+0}$

implify the expression to find the limit.
$\lim{x \rightarrow+\infty} \frac{4}{2} =2$ The limit of the function as $x$ approaches infinity is indeed2.