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# Limits of functions as x approaches infinity

As you compute limits of functions as x approaches a constant, you can find several possible solutions:

• The limit is equal to a constant
• The limit is plus or minus infinity
• The limit does not exit

We have three rules for evaluating the limit of a rational expression as x approaches infinity:

• If the highest power of x in a rational expression is in the numerator, then the limit as x approaches infinity is infinity.

• $\lim_{x\to\infty}\frac{5x^7-3x}{4x^6-3x^2}=\infty$

• If the highest power of x in a rational expression is in the denominator, then the limit as x approaches infinity is zero.

• $\lim_{x\to\infty}\frac{2x^5-3x}{4x^6-3x^2}=0$

• If the highest power of x in a rational expression is the same in both the numerator and denominator, then the limit as x approaches infinity is the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator.

• $\lim_{x\to\infty}\frac{5x^7-3x}{4x^7-3x^2}=\frac{5}{4}$

Compute:
$\lim_{\fs1x\rightarrow\infty}\;\frac{-8x^6+6x^4-2x^2-5}{8x^6-6x^4+7x+1}$
 Solution: (Only one is true) $-\infty$ $+\infty$ The number