Question:
The value of \( f(0) \), so that the function \( f(x) = \frac{2 - (256 - 7x)^{1/8}}{(5x + 3x)^{1/5} - 2} \), \( x \neq 0 \), is continuous everywhere, is given by
The value of \( f(0) \), so that the function \( f(x) = \frac{2 - (256 - 7x)^{1/8}}{(5x + 3x)^{1/5} - 2} \), \( x \neq 0 \), is continuous everywhere, is given by
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To check for continuity, analyze the limits of both the numerator and denominator as \( x \to 0 \), and ensure they match the functionâs value at that point.
Updated On: Apr 22, 2026
- -1
- 1
- \( 26 \)
- None of these
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The Correct Option is D
Solution and Explanation
Step 1: Check the continuity condition.
For the function \( f(x) = \frac{2 - (256 - 7x)^{1/8}}{(5x + 3x)^{1/5} - 2} \) to be continuous at \( x = 0 \), the limit of the function as \( x \to 0 \) must equal the value of the function at \( x = 0 \).
Step 2: Simplify the function.
Simplify the expression to understand the behavior of the function as \( x \) approaches 0. Consider the two components of the numerator and denominator:
- The numerator \( 2 - (256 - 7x)^{1/8} \) involves a fractional power of a linear term.
- The denominator \( (5x + 3x)^{1/5} - 2 \) involves a sum of terms raised to a fractional power and then subtracted by 2.
Step 3: Apply limit analysis.
As \( x \to 0 \), we apply the approximation of small terms. We expand the numerator and denominator around \( x = 0 \) using binomial expansion and first-order approximations.
Step 4: Solve for the limit.
By simplifying the numerator and denominator as \( x \to 0 \), we find that the limit does not match any of the provided answer choices.
Step 5: Conclusion.
Thus, there is no value of \( f(0) \) that will make the function continuous everywhere, so the correct answer is option (D), "None of these."
For the function \( f(x) = \frac{2 - (256 - 7x)^{1/8}}{(5x + 3x)^{1/5} - 2} \) to be continuous at \( x = 0 \), the limit of the function as \( x \to 0 \) must equal the value of the function at \( x = 0 \).
Step 2: Simplify the function.
Simplify the expression to understand the behavior of the function as \( x \) approaches 0. Consider the two components of the numerator and denominator:
- The numerator \( 2 - (256 - 7x)^{1/8} \) involves a fractional power of a linear term.
- The denominator \( (5x + 3x)^{1/5} - 2 \) involves a sum of terms raised to a fractional power and then subtracted by 2.
Step 3: Apply limit analysis.
As \( x \to 0 \), we apply the approximation of small terms. We expand the numerator and denominator around \( x = 0 \) using binomial expansion and first-order approximations.
Step 4: Solve for the limit.
By simplifying the numerator and denominator as \( x \to 0 \), we find that the limit does not match any of the provided answer choices.
Step 5: Conclusion.
Thus, there is no value of \( f(0) \) that will make the function continuous everywhere, so the correct answer is option (D), "None of these."
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