Question Statement

Given the piecewise function: $f(x)=\{\begin{array}{ll}mx& ,x<4\\ x^2& ,x\ge 4\end{array}$

Find the value of $m$ such that $f(x)$ is continuous at $x=4$.

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Background and Explanation

For a function to be continuous at a point $x=a$, the left-hand limit (approaching from values less than $a$) must equal the right-hand limit (approaching from values greater than $a$). This ensures the function has no jump discontinuity at that point.

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Solution

To determine the value of $m$ that makes $f(x)$ continuous at $x=4$, we calculate both the left-hand and right-hand limits and set them equal.

Left Hand Limit

For $x<4$, the function is defined as $f(x)=mx$. Therefore:

$\begin{array}{rl}\underset{x\to 4^{-}}{\lim }f(x)& =\underset{x\to 4^{-}}{\lim }(mx)\\ & =m(4)\\ \underset{x\to 4^{-}}{\lim }f(x)& =4m\end{array}$

Right Hand Limit

For $x\ge 4$, the function is defined as $f(x)=x^2$. Therefore:

$\begin{array}{rl}\underset{x\to 4^{+}}{\lim }f(x)& =\underset{x\to 4^{+}}{\lim }\left(x^2\right)\\ & =(4{)}^2\\ \underset{x\to 4^{+}}{\lim }f(x)& =16\end{array}$

Applying Continuity Condition

Since $f(x)$ is continuous at $x=4$, the left-hand limit must equal the right-hand limit:

$\begin{array}{rl}\underset{x\to 4^{-}}{\lim }f(x)& =\underset{x\to 4^{+}}{\lim }f(x)\\ 4m& =16\\ m& =4\end{array}$

Therefore, $m=4$.

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Key Formulas or Methods Used

• Continuity Condition: A function $f(x)$ is continuous at $x=a$ if ${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=f(a)$
• Left-Hand Limit: ${\lim }_{x\to a^{-}}f(x)$ — the value approached as $x$ approaches $a$ from the left
• Right-Hand Limit: ${\lim }_{x\to a^{+}}f(x)$ — the value approached as $x$ approaches $a$ from the right
• Direct Substitution: For continuous elementary functions, ${\lim }_{x\to a}f(x)=f(a)$

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Summary of Steps

1. Identify the left piece: For $x<4$, use $f(x)=mx$ and calculate ${\lim }_{x\to 4^{-}}mx=4m$
2. Identify the right piece: For $x\ge 4$, use $f(x)=x^2$ and calculate ${\lim }_{x\to 4^{+}}{x}^2=16$
3. Set limits equal: For continuity, $4m=16$
4. Solve: Divide both sides by 4 to get $m=4$