Evaluate the limits. Enter DNE if the limit does not exist. a) $$ \lim _{x \rightarrow b} q(x)=\left\{\begin{array}{ll}a(x-b)^{3}+c & xb\end{array}\right. $$ b) $$ \lim _{x \rightarrow b^{+}} q(x)=\square $$ c) $$ \lim _{x \rightarrow b} q(x)=\square $$ d) $$ q(b)=\square $$

Question

UpStudy AI · Accepted Answer

a) Since for $$ xb $$ we have
$$
q(x)=a(x-b)+c,
$$
we compute the left-hand limit:
$$
\lim_{x \to b^-} q(x)=a\lim_{x \to b^-}(x-b)^3+c=a\cdot 0+c=c.
$$
Similarly, the right-hand limit is:
$$
\lim_{x \to b^+} q(x)=a\lim_{x \to b^+}(x-b)+c=a\cdot 0+c=c.
$$
Since both one-sided limits agree, we have:
$$
\lim_{x \to b} q(x)=c.
$$

b) For $$ x \to b^+ $$,
$$
\lim_{x \to b^+} q(x)=a\cdot 0+c=c.
$$

c) As shown in part (a), 
$$
\lim_{x \to b} q(x)=c.
$$

d) The function $$ q(x) $$ is defined by different expressions for $$ xb $$ but no value is given for $$ x=b $$. Therefore,
$$
q(b)=\text{DNE}.
$$
a) The limit as $$ x $$ approaches $$ b $$ from the left is $$ c $$. b) The limit as $$ x $$ approaches $$ b $$ from the right is $$ c $$. c) The overall limit as $$ x $$ approaches $$ b $$ is $$ c $$. d) The value of $$ q(b) $$ does not exist.

Evaluate the limits.
Enter DNE if the limit does not exist.
a)
b)
c)
d)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Calculus Questions

Evaluate the limits. Enter DNE if the limit does not exist. a) b) c) d)