Question Statement

Find the values of $m$ and $n$ such that the function $f(x)$ is continuous at $x=3$, where:

$f(x)=\{\begin{array}{ll}mx& ,x<3\\ n& ,x=3\\ -2x+9& ,x>3\end{array}$

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Background and Explanation

For a function to be continuous at a point $x=a$, the left-hand limit, right-hand limit, and the function value at that point must all be equal: $\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=f(a)$. We will apply this condition at $x=3$ to find the unknown constants.

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Solution

Finding the Left-Hand Limit

To find the limit as $x$ approaches $3$ from the left ($x<3$), we use the first piece of the function $f(x)=mx$:

${\lim }_{x\to 3^{-}}f(x)={\lim }_{x\to 3^{-}}(mx)$

Substituting $x=3$:

$=m(3)=3m$

Finding the Right-Hand Limit

To find the limit as $x$ approaches $3$ from the right ($x>3$), we use the third piece of the function $f(x)=-2x+9$:

${\lim }_{x\to 3^{+}}f(x)={\lim }_{x\to 3^{+}}(-2x+9)$

Substituting $x=3$:

$=-2(3)+9=-6+9=3$

Finding the Function Value at $x=3$

From the definition of the function, at $x=3$:

$f(3)=n$

Applying the Continuity Condition

Since $f(x)$ is continuous at $x=3$, the three values must be equal:

${\lim }_{x\to 3^{-}}f(x)={\lim }_{x\to 3^{+}}f(x)=f(3)$

Substituting the values we found:

$3m=3=n$

Solving for $m$ and $n$:

• From $3m=3$, we get $\boxed{m=1}$
• From $3=n$, we get $\boxed{n=3}$

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Key Formulas or Methods Used

• Continuity Condition: $\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=f(a)$
• Left-Hand Limit: $\underset{x\to 3^{-}}{\lim }f(x)=\underset{x\to 3^{-}}{\lim }(mx)=3m$
• Right-Hand Limit: $\underset{x\to 3^{+}}{\lim }f(x)=\underset{x\to 3^{+}}{\lim }(-2x+9)=3$
• Function Value: $f(3)=n$

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Summary of Steps

1. Calculate Left-Hand Limit: Use $f(x)=mx$ for $x<3$ to get ${\lim }_{x\to 3^{-}}f(x)=3m$
2. Calculate Right-Hand Limit: Use $f(x)=-2x+9$ for $x>3$ to get ${\lim }_{x\to 3^{+}}f(x)=3$
3. Identify Function Value: From the middle piece, $f(3)=n$
4. Set Equal for Continuity: Equate $3m=3=n$
5. Solve: $m=1$ and $n=3$