evaluate the limit by rationalizing the numerator. lim(x→ - 3) (√(10x + 66) - √(15x + 81))/(-5x - 15)…

Answer:

$\frac{1}{2}$

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $\sqrt{10x + 66}+\sqrt{15x + 81}$ [ \frac{(\sqrt{10x + 66}-\sqrt{15x + 81})(\sqrt{10x + 66}+\sqrt{15x + 81})}{(-5x - 15)(\sqrt{10x + 66}+\sqrt{15x + 81})} ]

Step2: Expand numerator

Use $(a - b)(a + b)=a^{2}-b^{2}$ in numerator. [ \frac{(10x + 66)-(15x + 81)}{(-5x - 15)(\sqrt{10x + 66}+\sqrt{15x + 81})} ] [ =\frac{10x + 66-15x - 81}{(-5x - 15)(\sqrt{10x + 66}+\sqrt{15x + 81})} ] [ =\frac{-5x - 15}{(-5x - 15)(\sqrt{10x + 66}+\sqrt{15x + 81})} ]

Step3: Simplify fraction

Cancel out $-5x - 15$ (for $x\neq - 3$) [ \frac{1}{\sqrt{10x + 66}+\sqrt{15x + 81}} ]

Step4: Evaluate limit

Substitute $x=-3$ [ \frac{1}{\sqrt{10(-3)+66}+\sqrt{15(-3)+81}}=\frac{1}{\sqrt{36}+\sqrt{36}}=\frac{1}{6 + 6}=\frac{1}{2} ]