Question

evaluate the limit.
lim _ { x
ightarrow infty } \frac { - 9 x - 7 } { - 5 x ^ { 2 } + 5 x - 3 }
simplify any fractions in your answer.

Explanation:

Step1: Divide numerator and denominator by \(x^{2}\)

$$\lim_{x ightarrow\infty}\frac{\frac{-9x}{x^{2}}-\frac{7}{x^{2}}}{\frac{-5x^{2}}{x^{2}}+\frac{5x}{x^{2}}-\frac{3}{x^{2}}}$$

Step2: Simplify the fractions

$$\lim_{x ightarrow\infty}\frac{-\frac{9}{x}-\frac{7}{x^{2}}}{-5 + \frac{5}{x}-\frac{3}{x^{2}}}$$

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

ightarrow\infty}\frac{1}{x^{n}} = 0\) (\(n>0\))
As \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}\frac{9}{x}=0\), \(\lim_{x
ightarrow\infty}\frac{7}{x^{2}} = 0\), \(\lim_{x
ightarrow\infty}\frac{5}{x}=0\), \(\lim_{x
ightarrow\infty}\frac{3}{x^{2}}=0\)

Substitute these values into the expression:
$$\frac{0 - 0}{-5+0 - 0}$$

Answer:

\(0\)