Question

suppose that \\(\lim\limits_{x\to -4} p(x) = 8\\), \\(\lim\limits_{x\to -4} r(x) = 0\\), and \\(\lim\limits_{x\to -4} s(x) = -9\\). find the limits in parts (a) through (c) below.
a. \\(\lim\limits_{x\to -4} (p(x) + r(x) + s(x)) = \boxed{16}\\)
(simplify your answer.)

Explanation:

Step1: Apply Sum Rule of Limits

The sum rule for limits states that \(\lim_{x \to a} (f(x) + g(x) + h(x))=\lim_{x \to a} f(x)+\lim_{x \to a} g(x)+\lim_{x \to a} h(x)\). So for \(\lim_{x \to - 4}(p(x)+r(x)+s(x))\), we can split it as \(\lim_{x \to - 4}p(x)+\lim_{x \to - 4}r(x)+\lim_{x \to - 4}s(x)\).

Step2: Substitute the Given Limits

We know that \(\lim_{x \to - 4}p(x) = 8\), \(\lim_{x \to - 4}r(x)=0\) and \(\lim_{x \to - 4}s(x)=- 9\). Substituting these values into the expression from Step 1, we get \(8 + 0+( - 9)\).

Step3: Simplify the Expression

Calculate \(8+0 - 9=8 - 9=-1\). Wait, the original answer in the problem was 16, which seems to be incorrect. Let's re - check. Wait, maybe there is a mis - reading. Wait, \(\lim_{x
ightarrow - 4}p(x) = 8\), \(\lim_{x
ightarrow - 4}r(x)=0\), \(\lim_{x
ightarrow - 4}s(x)=-9\). Then \(8 + 0+( - 9)=-1\). But if we assume that maybe the value of \(\lim_{x
ightarrow - 4}s(x)\) is 9 instead of - 9, then \(8 + 0+9 = 17\), still not 16. Wait, maybe the problem has a typo, but according to the correct limit laws:

The sum rule of limits: If \(\lim_{x
ightarrow a}f(x)=L_1\), \(\lim_{x
ightarrow a}g(x)=L_2\) and \(\lim_{x
ightarrow a}h(x)=L_3\), then \(\lim_{x
ightarrow a}(f(x)+g(x)+h(x))=L_1 + L_2+L_3\)

So with \(L_1 = 8\), \(L_2 = 0\), \(L_3=-9\), the correct value is \(8 + 0+( - 9)=-1\)

But if we follow the wrong calculation in the problem (maybe a typo in the given values), if we assume that \(\lim_{x
ightarrow - 4}s(x)=8\) (instead of - 9), then \(8 + 0+8 = 16\). But based on the given values, the correct calculation is:

\(\lim_{x
ightarrow - 4}(p(x)+r(x)+s(x))=\lim_{x
ightarrow - 4}p(x)+\lim_{x
ightarrow - 4}r(x)+\lim_{x
ightarrow - 4}s(x)=8 + 0+( - 9)=-1\)

Answer:

\(-1\) (If we follow the given limit values. If the answer 16 is expected, there might be a typo in the problem's limit values)