Question

homework 1.2: limits involving infinity
score: 17/36 answered: 11/18
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question 12
0/2 pts 8 49 details
evaluate
lim(x→9) (x² - 81)/(x - 9) =
(if the limit does not exist, type dne.)

Explanation:

Step1: Factor the numerator

We know that \(x^{2}-81=(x + 9)(x - 9)\) by the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\) where \(a=x\) and \(b = 9\). So the limit becomes \(\lim_{x
ightarrow9}\frac{(x + 9)(x - 9)}{x - 9}\).

Step2: Simplify the function

Since \(x
eq9\) when taking the limit (we are approaching 9, not equal to 9), we can cancel out the common factor \((x - 9)\) in the numerator and the denominator. The function simplifies to \(\lim_{x
ightarrow9}(x + 9)\).

Step3: Evaluate the limit

Substitute \(x = 9\) into \(x+9\). We get \(9 + 9=18\).

Answer:

18