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Evaluate the following limits:
[tex]\[
\lim _{x \rightarrow \infty} f(x) \quad \text{and} \quad \lim _{x \rightarrow -\infty} f(x).
\][/tex]

Given:
[tex]\[
f(x) = \frac{-3 e^{\frac{1}{x}} + 4}{-4 e^{\frac{1}{x}} + 6}
\][/tex]

If the function increases without bound, enter [tex]$\infty$[/tex]. If the function decreases without bound, enter [tex]$-\infty$[/tex]. If the function does not approach a finite limit nor [tex]$\pm \infty$[/tex] as [tex]$x \rightarrow \pm \infty$[/tex], enter [tex]$\varnothing$[/tex].



Answer :

GinnyAnswer
Certainly! Let's evaluate the given limits step by step:

We are given:
[tex]\[ f(x) = \frac{-3 \exp\left(\frac{1}{x}\right) + 4}{-4 \exp\left(\frac{1}{x}\right) + 6} \][/tex]

### Evaluating [tex]\(\lim_{x \rightarrow \infty} f(x)\)[/tex]:

1. Behavior of [tex]\(\exp\left(\frac{1}{x}\right)\)[/tex] as [tex]\(x \rightarrow \infty\)[/tex]:

As [tex]\(x\)[/tex] approaches infinity, [tex]\(\frac{1}{x}\)[/tex] approaches 0. Therefore, [tex]\(\exp\left(\frac{1}{x}\right)\)[/tex] approaches [tex]\(\exp(0)\)[/tex], which is 1.

2. Substitute [tex]\(\exp\left(\frac{1}{x}\right) \approx 1\)[/tex] in [tex]\(f(x)\)[/tex] as [tex]\(x \to \infty\)[/tex]:
[tex]\[ f(x) \approx \frac{-3 \cdot 1 + 4}{-4 \cdot 1 + 6} \][/tex]

Simplify the expression:
[tex]\[ f(x) \approx \frac{-3 + 4}{-4 + 6} = \frac{1}{2} \][/tex]

Thus,
[tex]\[ \lim_{x \rightarrow \infty} f(x) = \frac{1}{2} \][/tex]

### Evaluating [tex]\(\lim_{x \rightarrow -\infty} f(x)\)[/tex]:

3. Behavior of [tex]\(\exp\left(\frac{1}{x}\right)\)[/tex] as [tex]\(x \rightarrow -\infty\)[/tex]:

As [tex]\(x\)[/tex] approaches negative infinity, [tex]\(\frac{1}{x}\)[/tex] again approaches 0. Therefore, [tex]\(\exp\left(\frac{1}{x}\right)\)[/tex] approaches [tex]\(\exp(0)\)[/tex], which is also 1.

4. Substitute [tex]\(\exp\left(\frac{1}{x}\right) \approx 1\)[/tex] in [tex]\(f(x)\)[/tex] as [tex]\(x \to -\infty\)[/tex]:
[tex]\[ f(x) \approx \frac{-3 \cdot 1 + 4}{-4 \cdot 1 + 6} \][/tex]

Simplify the expression:
[tex]\[ f(x) \approx \frac{-3 + 4}{-4 + 6} = \frac{1}{2} \][/tex]

Thus,
[tex]\[ \lim_{x \rightarrow -\infty} f(x) = \frac{1}{2} \][/tex]

### Conclusion
The limits are:
[tex]\[ \boxed{ \left(\frac{1}{2}, \frac{1}{2}\right) } \][/tex]

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