lim(x→9) (9 - x)/(3 - √x)

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the denominator
Multiply the numerator and denominator by the conjugate of the denominator $3 + \sqrt{x}$.
$$
\begin{align*}
\lim_{x
ightarrow9}\frac{9 - x}{3-\sqrt{x}}&=\lim_{x
ightarrow9}\frac{(9 - x)(3+\sqrt{x})}{(3-\sqrt{x})(3+\sqrt{x})}\
\end{align*}
$$
## Step2: Simplify the denominator
Use the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$. Here $a = 3$ and $b=\sqrt{x}$, so $(3-\sqrt{x})(3+\sqrt{x})=9 - x$.
$$
\begin{align*}
\lim_{x
ightarrow9}\frac{(9 - x)(3+\sqrt{x})}{(3-\sqrt{x})(3+\sqrt{x})}&=\lim_{x
ightarrow9}\frac{(9 - x)(3+\sqrt{x})}{9 - x}\
\end{align*}
$$
## Step3: Cancel out the common factor
Cancel out the common factor $(9 - x)$ (since $x
eq9$ when taking the limit).
$$
\begin{align*}
\lim_{x
ightarrow9}\frac{(9 - x)(3+\sqrt{x})}{9 - x}&=\lim_{x
ightarrow9}(3+\sqrt{x})
\end{align*}
$$
## Step4: Evaluate the limit
Substitute $x = 9$ into $3+\sqrt{x}$.
$$
\begin{align*}
\lim_{x
ightarrow9}(3+\sqrt{x})&=3+\sqrt{9}\
&=3 + 3\
&=6
\end{align*}
$$
# Answer:
$6$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the denominator

Step2: Simplify the denominator

Step3: Cancel out the common factor

Step4: Evaluate the limit

Answer: