mit: \\( \\lim _ { x \\rightarrow 64 } \\frac { - 128 + 2 x } { 8 - \\sqrt { x } } \\)
mit: \\( \\lim _ { x \\rightarrow 64 } \\frac { - 128 + 2 x } { 8 - \\sqrt { x } } \\)
Answer
Explanation:
Step1: Factor numerator
Factor out (2) from (-128 + 2x): (2(x - 64)). So, the limit becomes (\lim_{x\rightarrow64}\frac{2(x - 64)}{8-\sqrt{x}}).
Step2: Rationalize denominator
Multiply numerator and denominator by (8+\sqrt{x}): [ \begin{align*} &\lim_{x\rightarrow64}\frac{2(x - 64)(8+\sqrt{x})}{(8-\sqrt{x})(8+\sqrt{x})}\ =&\lim_{x\rightarrow64}\frac{2(x - 64)(8+\sqrt{x})}{64 - x}\ =&\lim_{x\rightarrow64}\frac{-2(64 - x)(8+\sqrt{x})}{64 - x}\ \end{align*} ]
Step3: Cancel out common factor
Cancel out ((64 - x)): (\lim_{x\rightarrow64}-2(8+\sqrt{x}))
Step4: Substitute (x = 64)
Substitute (x = 64) into (-2(8+\sqrt{x})): (-2(8+\sqrt{64})=-2(8 + 8)=-32)
Answer:
(-32)