Find $$ \lim _{x \rightarrow \infty} f(x) $$ if $$ \frac{4 x-1}{x}5 $$

Question

UpStudy AI · Accepted Answer

Given the inequality

$$
\frac{4x-1}{x} < f(x) < \frac{4x^2+3x}{x^2} \quad \text{for all } x > 5,
$$

first simplify the left-hand side:

$$
\frac{4x-1}{x} = 4 - \frac{1}{x}.
$$

Next, simplify the right-hand side:

$$
\frac{4x^2+3x}{x^2} = 4 + \frac{3}{x}.
$$

Thus, the inequality becomes

$$
4 - \frac{1}{x} < f(x) < 4 + \frac{3}{x}.
$$

Now, take the limit as $$ x \rightarrow \infty $$ for both sides:

For the left-hand side:

$$
\lim_{x \to \infty} \left(4 - \frac{1}{x}\right) = 4 - 0 = 4.
$$

For the right-hand side:

$$
\lim_{x \to \infty} \left(4 + \frac{3}{x}\right) = 4 + 0 = 4.
$$

Since $$ f(x) $$ is squeezed between two functions that both approach $$ 4 $$, by the Squeeze Theorem,

$$
\lim_{x \to \infty} f(x) = 4.
$$
The limit of $$ f(x) $$ as $$ x $$ approaches infinity is 4.

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