QuestionAugust 24, 2025

If fis the function defined by f(x)=(x-9)/(sqrt (x)-3) then lim _(xarrow 9)f(x) is equivalent to which of the following? A lim _(xarrow 9)(sqrt (x)-3) B lim _(xarrow 9)(sqrt (x)+3) C lim _(xarrow 9)((x^2-81)/(x-9)) D (lim _(xarrow 9)(x-9))/(lim _(xarrow 9)(sqrt (x)-3))

If fis the function defined by f(x)=(x-9)/(sqrt (x)-3) then lim _(xarrow 9)f(x) is equivalent to which of the following? A lim _(xarrow 9)(sqrt (x)-3) B lim _(xarrow 9)(sqrt (x)+3) C lim _(xarrow 9)((x^2-81)/(x-9)) D (lim _(xarrow 9)(x-9))/(lim _(xarrow 9)(sqrt (x)-3))
If fis the function defined by f(x)=(x-9)/(sqrt (x)-3) then lim _(xarrow 9)f(x) is equivalent to which of the following?
A
lim _(xarrow 9)(sqrt (x)-3)
B
lim _(xarrow 9)(sqrt (x)+3)
C
lim _(xarrow 9)((x^2-81)/(x-9))
D
(lim _(xarrow 9)(x-9))/(lim _(xarrow 9)(sqrt (x)-3))

Solution
4.3(210 votes)

Answer

A: \lim _{x\rightarrow 9}(\sqrt {x}-3) Explanation 1. Simplify the function Factor the numerator x-9 as (\sqrt{x}-3)(\sqrt{x}+3). 2. Cancel common terms Cancel (\sqrt{x}-3) in the numerator and denominator, simplifying f(x) to \sqrt{x}+3. 3. Evaluate the limit Substitute x = 9 into the simplified function: \lim_{x \rightarrow 9} (\sqrt{x} + 3) = \sqrt{9} + 3 = 6.

Explanation

1. Simplify the function<br /> Factor the numerator $x-9$ as $(\sqrt{x}-3)(\sqrt{x}+3)$.<br />2. Cancel common terms<br /> Cancel $(\sqrt{x}-3)$ in the numerator and denominator, simplifying $f(x)$ to $\sqrt{x}+3$.<br />3. Evaluate the limit<br /> Substitute $x = 9$ into the simplified function: $\lim_{x \rightarrow 9} (\sqrt{x} + 3) = \sqrt{9} + 3 = 6$.
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