Question:
$\lim_{x \to 3} \frac{(84-x)^{\frac{1}{4}} - 3}{x - 3}$ is
$\lim_{x \to 3} \frac{(84-x)^{\frac{1}{4}} - 3}{x - 3}$ is
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Use derivative form of limit when expression matches pattern.
Updated On: Apr 26, 2026
- $\frac{-1}{108}$
- $\frac{-1}{84}$
- $\frac{-1}{27}$
- $\frac{-1}{4}$
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The Correct Option is A
Solution and Explanation
Concept:
\[ \lim_{x \to a} \frac{f(x)-f(a)}{x-a} = f'(a) \] Step 1: Define function. \[ f(x) = (84-x)^{1/4} \]
Step 2: Differentiate. \[ f'(x) = \frac{-1}{4(84-x)^{3/4}} \]
Step 3: Substitute $x=3$. \[ = \frac{-1}{4(81)^{3/4}} = \frac{-1}{4 \cdot 27} \]
Step 4: Conclusion. \[ = \frac{-1}{108} \]
\[ \lim_{x \to a} \frac{f(x)-f(a)}{x-a} = f'(a) \] Step 1: Define function. \[ f(x) = (84-x)^{1/4} \]
Step 2: Differentiate. \[ f'(x) = \frac{-1}{4(84-x)^{3/4}} \]
Step 3: Substitute $x=3$. \[ = \frac{-1}{4(81)^{3/4}} = \frac{-1}{4 \cdot 27} \]
Step 4: Conclusion. \[ = \frac{-1}{108} \]
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