question evaluate the limit: $lim_{x\rightarrow25}\frac{50 - 2x}{5-sqrt{x}}$
question evaluate the limit: $lim_{x\rightarrow25}\frac{50 - 2x}{5-sqrt{x}}$
Answer
Answer:
$10$
Explanation:
Step1: Rationalize the denominator
Multiply numerator and denominator by $5 + \sqrt{x}$: [ \begin{align*} \lim_{x\rightarrow25}\frac{50 - 2x}{5-\sqrt{x}}\times\frac{5+\sqrt{x}}{5+\sqrt{x}}&=\lim_{x\rightarrow25}\frac{(50 - 2x)(5+\sqrt{x})}{25 - x}\ \end{align*} ]
Step2: Factor out - 2 from the numerator
Factor $50 - 2x=-2(x - 25)$: [ \begin{align*} \lim_{x\rightarrow25}\frac{-2(x - 25)(5+\sqrt{x})}{25 - x}&=\lim_{x\rightarrow25}\frac{-2(x - 25)(5+\sqrt{x})}{-(x - 25)}\ \end{align*} ]
Step3: Cancel out the common factor
Cancel out $(x - 25)$: [ \begin{align*} \lim_{x\rightarrow25}\frac{-2(x - 25)(5+\sqrt{x})}{-(x - 25)}&=\lim_{x\rightarrow25}2(5+\sqrt{x}) \end{align*} ]
Step4: Substitute the value of x
Substitute $x = 25$: [ \begin{align*} 2(5+\sqrt{25})&=2(5 + 5)\ &=2\times10=10 \end{align*} ]