find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise…

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)\n\\( \\lim _ { x \\rightarrow \\infty } \\left( \\frac { x - 9 } { x ^ { 2 } + 7 } \\right) \\)

find the limit. (if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)\n\\( \\lim _ { x \\rightarrow \\infty } \\left( \\frac { x - 9 } { x ^ { 2 } + 7 } \\right) \\)

Answer

Explanation:

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

$$\lim_{x\rightarrow\infty}\frac{\frac{x - 9}{x^{2}}}{\frac{x^{2}+7}{x^{2}}}=\lim_{x\rightarrow\infty}\frac{\frac{x}{x^{2}}-\frac{9}{x^{2}}}{\frac{x^{2}}{x^{2}}+\frac{7}{x^{2}}}$$

Step2: Simplify the fractions

$$=\lim_{x\rightarrow\infty}\frac{\frac{1}{x}-\frac{9}{x^{2}}}{1 + \frac{7}{x^{2}}}$$

Step3: Use the limit property (\lim_{x\rightarrow\infty}\frac{1}{x^{n}} = 0) for (n>0)

As (x\rightarrow\infty), (\lim_{x\rightarrow\infty}\frac{1}{x}=0), (\lim_{x\rightarrow\infty}\frac{9}{x^{2}} = 0) and (\lim_{x\rightarrow\infty}\frac{7}{x^{2}}=0) $$\frac{0 - 0}{1+0}$$

Answer:

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