Question

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)
\\( \lim _ { x \
ightarrow \infty } \frac { 9 x - 2 } { 2 x + 3 } \\)

Explanation:

Step1: Divide numerator and denominator by \(x\)

$$\lim_{x ightarrow\infty}\frac{9x - 2}{2x + 3}=\lim_{x ightarrow\infty}\frac{\frac{9x}{x}-\frac{2}{x}}{\frac{2x}{x}+\frac{3}{x}}$$

Step2: Simplify the expression

$$=\lim_{x ightarrow\infty}\frac{9-\frac{2}{x}}{2+\frac{3}{x}}$$

Step3: Use the limit property \(\lim_{x

ightarrow\infty}\frac{c}{x}=0\) (\(c\) is a constant)
As \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}\frac{2}{x}=0\) and \(\lim_{x
ightarrow\infty}\frac{3}{x}=0\). So the limit becomes \(\frac{9 - 0}{2+0}\)

Answer:

\(\frac{9}{2}\)