b) determine whether each function is continuous or
discontinuous at the given-value. Examine the three
.conditions in the definition of continuity

Ask by Griffiths Beck. in Saudi Arabia

Mar 23,2025

Question

b) determine whether each function is continuous or
discontinuous at the given-value. Examine the three
.conditions in the definition of continuity

Ask by Griffiths Beck. in Saudi Arabia

Mar 23,2025

UpStudy AI · Accepted Answer

1) For 
$$
f(x)=\begin{cases}
3x-2, & x \geq 3\[5mm]
x+4, & x < 3
\end{cases}
$$
at $$ x = 3 $$:

- **Step 1:** Evaluate $$ f(3) $$.  
  Since $$ x=3 $$ falls in the region $$ x \geq 3 $$,
  $$
  f(3)=3(3)-2=9-2=7.
  $$

- **Step 2:** Find the right-hand limit as $$ x \to 3^+ $$.  
  For $$ x>3 $$, we use $$ 3x-2 $$. Thus,
  $$
  \lim_{x \to 3^+} f(x)=3(3)-2=9-2=7.
  $$

- **Step 3:** Find the left-hand limit as $$ x \to 3^- $$.  
  For $$ x<3 $$, we use $$ x+4 $$. Thus,
  $$
  \lim_{x \to 3^-} f(x)=3+4=7.
  $$

- **Conclusion:** Since $$ f(3) $$ exists, the left- and right-hand limits exist and are equal to $$ 7 $$, the function is continuous at $$ x=3 $$.

2) For 
$$
f(x)=\frac{x+4}{x-3}
$$
at $$ x=-2 $$:

- **Step 1:** Confirm that $$ f(-2) $$ is defined.  
  We calculate the denominator at $$ x=-2 $$:
  $$
  -2-3=-5 \quad (\neq 0),
  $$
  therefore,
  $$
  f(-2)=\frac{-2+4}{-2-3}=\frac{2}{-5}=-\frac{2}{5}.
  $$

- **Step 2:** Since $$ f(x) $$ is a rational function and its denominator is nonzero at $$ x=-2 $$, the function is continuous at $$ x=-2 $$ with
  $$
  \lim_{x \to -2} f(x)=f(-2)=-\frac{2}{5}.
  $$

- **Conclusion:** The function is continuous at $$ x=-2 $$.
1) The function $$ f(x) $$ is continuous at $$ x=3 $$. 2) The function $$ f(x) $$ is continuous at $$ x=-2 $$.

b) determine whether each function is continuous or
discontinuous at the given-value. Examine the three
.conditions in the definition of continuity

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b) determine whether each function is continuous or discontinuous at the given-value. Examine the three .conditions in the definition of continuity

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b) determine whether each function is continuous or
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