Question

it:

$$ lim _ { x ightarrow - 15 ^ { + } } \frac { 2 x + 30 } { sqrt { x + 15 } } $$

Explanation:

Step1: Factor the numerator

Factor out \(2\) from \(2x + 30\), we get \(2(x + 15)\). So the limit becomes \(\lim_{x
ightarrow - 15^{+}}\frac{2(x + 15)}{\sqrt{x + 15}}\).

Step2: Simplify the expression

Let \(t=x + 15\). As \(x
ightarrow - 15^{+}\), \(t
ightarrow0^{+}\). The expression \(\frac{2(x + 15)}{\sqrt{x + 15}}=\frac{2t}{\sqrt{t}}\) (\(t>0\)). Since \(\frac{2t}{\sqrt{t}}=2\sqrt{t}\) for \(t>0\).

Step3: Evaluate the limit

Now we find \(\lim_{t
ightarrow0^{+}}2\sqrt{t}\). Substitute \(t = 0\) into \(2\sqrt{t}\), we know that \(\lim_{t
ightarrow0^{+}}2\sqrt{t}=0\).

Answer:

\(0\)