if k(x) = (x³ + 343)/(x + 7), complete the table and use the results to find lim x→ - 7 k(x). x - 7.1 - 7.01…

Explanation:

Step1: Factor the numerator

We know that (a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})), so (x^{3}+343=x^{3}+7^{3}=(x + 7)(x^{2}-7x + 49)). Then (k(x)=\frac{(x + 7)(x^{2}-7x + 49)}{x + 7}=x^{2}-7x + 49) for (x\neq - 7).

Step2: Calculate (k(x)) for each (x) - value

When (x=-7.1)

Substitute (x=-7.1) into (y=x^{2}-7x + 49). (y=(-7.1)^{2}-7\times(-7.1)+49=50.41 + 49.7+49 = 149.110)

When (x=-7.01)

Substitute (x=-7.01) into (y=x^{2}-7x + 49). (y=(-7.01)^{2}-7\times(-7.01)+49=49.1401+49.07 + 49=147.210)

When (x=-7.001)

Substitute (x=-7.001) into (y=x^{2}-7x + 49). (y=(-7.001)^{2}-7\times(-7.001)+49=49.0014001+49.007+49 = 147.008)

When (x=-6.999)

Substitute (x=-6.999) into (y=x^{2}-7x + 49). (y=(-6.999)^{2}-7\times(-6.999)+49=48.986001+48.993+49=146.979)

When (x=-6.99)

Substitute (x=-6.99) into (y=x^{2}-7x + 49). (y=(-6.99)^{2}-7\times(-6.99)+49=48.8601+48.93+49 = 146.790)

When (x=-6.9)

Substitute (x=-6.9) into (y=x^{2}-7x + 49). (y=(-6.9)^{2}-7\times(-6.9)+49=47.61+48.3+49=144.910)

Step3: Find the limit

As (x\to - 7), we can use the simplified function (k(x)=x^{2}-7x + 49). (\lim_{x\to - 7}k(x)=\lim_{x\to - 7}(x^{2}-7x + 49)=(-7)^{2}-7\times(-7)+49=49 + 49+49=147)

Answer:

(x)	(k(x))
(-7.1)	(149.110)
(-7.01)	(147.210)
(-7.001)	(147.008)
(-6.999)	(146.979)
(-6.99)	(146.790)
(-6.9)	(144.910)
(\lim_{x\to - 7}k(x)=147)