question\nevaluate the limit: $$ lim _ { x \rightarrow - 15 ^ { + } } \frac { - 4 x - 60 } { sqrt { x + 15 }…

question\nevaluate the limit: $$ lim _ { x \rightarrow - 15 ^ { + } } \frac { - 4 x - 60 } { sqrt { x + 15 } } $$

question\nevaluate the limit: $$ lim _ { x \rightarrow - 15 ^ { + } } \frac { - 4 x - 60 } { sqrt { x + 15 } } $$

Answer

Explanation:

Step1: Factor numerator

Factor out (-4) from (-4x - 60), we get (-4(x + 15)). So the limit becomes (\lim_{x\rightarrow - 15^{+}}\frac{-4(x + 15)}{\sqrt{x + 15}}).

Step2: Simplify the expression

Let (t=\sqrt{x + 15}), then (x+15=t^{2}) and as (x\rightarrow - 15^{+}), (t\rightarrow0^{+}). The expression (\frac{-4(x + 15)}{\sqrt{x + 15}}=\frac{-4t^{2}}{t}=-4t) for (t>0).

Step3: Evaluate the limit

Now we find (\lim_{t\rightarrow0^{+}}(-4t)). Substitute (t = 0) into (-4t).

Answer:

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