Math / Calculus

Question16. Determine the value of $f(x)$ when $x \rightarrow-\infty$ if $f(x)=\frac{3 x+2}{2 x+3}$ a) $-\frac{2}{3}$ b) $\frac{2}{3}$ c) $1.5^{+}$ d) $1.5^{-}$

Studdy Solution

STEP 1

1. We are given the function $f(x) = \frac{3x + 2}{2x + 3}$ .
2. We need to determine the value of $f(x)$ as $x \rightarrow -\infty$ .

STEP 2

1. Analyze the behavior of the function as $x \rightarrow -\infty$ .
2. Simplify the function by dividing numerator and denominator by $x$ .
3. Evaluate the limit of the simplified function as $x \rightarrow -\infty$ .

STEP 3

Analyze the behavior of the function as $x \rightarrow -\infty$ :
As $x$ becomes very large in magnitude and negative, the terms $3x$ and $2x$ will dominate over the constants $2$ and $3$ respectively.

STEP 4

Simplify the function by dividing both the numerator and the denominator by $x$ :
$f(x) = \frac{3x + 2}{2x + 3}$
Divide each term by $x$ :
$f(x) = \frac{\frac{3x}{x} + \frac{2}{x}}{\frac{2x}{x} + \frac{3}{x}}$
Simplify:
$f(x) = \frac{3 + \frac{2}{x}}{2 + \frac{3}{x}}$

STEP 5

Evaluate the limit of the simplified function as $x \rightarrow -\infty$ :
As $x \rightarrow -\infty$ , the terms $\frac{2}{x}$ and $\frac{3}{x}$ approach zero:
$\lim_{x \to -\infty} f(x) = \frac{3 + 0}{2 + 0} = \frac{3}{2}$
Since the limit is positive and constant, it is not affected by the sign of $x$ .
The value of $f(x)$ as $x \rightarrow -\infty$ is:
$\boxed{\frac{3}{2}}$
However, none of the given options directly match this result, which suggests a possible error in the options provided. The closest interpretation might be $1.5^{+}$ or $1.5^{-}$ , but typically these denote limits approaching from the positive or negative side of a point, not at infinity.

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