Question

find the following limit or state that it does not exist. assume b is a fixed real number.
lim_{x
ightarrow b}\frac{(x - b)^{30}-3x + 3b}{x - b}
simplify the given limit.
lim_{x
ightarrow b}\frac{(x - b)^{30}-3x + 3b}{x - b}=lim_{x
ightarrow b}(square)\text{ (simplify your answer.)}

Explanation:

Step1: Rewrite the numerator

We can rewrite $-3x + 3b$ as $-3(x - b)$. So the limit $\lim_{x
ightarrow b}\frac{(x - b)^{30}-3x + 3b}{x - b}=\lim_{x
ightarrow b}\frac{(x - b)^{30}-3(x - b)}{x - b}$.

Step2: Factor out common factor

Factor out $(x - b)$ from the numerator: $\lim_{x
ightarrow b}\frac{(x - b)[(x - b)^{29}-3]}{x - b}$.

Step3: Cancel out common factor

Cancel out the common factor $(x - b)$ in the numerator and the denominator. We get $\lim_{x
ightarrow b}[(x - b)^{29}-3]$.

Step4: Substitute $x = b$

Substitute $x = b$ into the expression $(x - b)^{29}-3$. Since $x - b$ becomes $b - b=0$, we have $(b - b)^{29}-3=0 - 3=-3$.

Answer:

$-3$