Given [tex]$\lim _{x \rightarrow 0} f(x)=4$[/tex]. What is [tex]$\lim _{x \rightarrow 0} \frac{1}{4}[f(x)]^4$[/tex]?
A. 0
B. 4
C. 64
D. 128
Answer (2)
We use limit properties to find that lim x → 0 4 1 [ f ( x ) ] 4 = 4 1 ⋅ 256 = 64 . Thus, the answer is 64, which is option C.
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Apply the limit properties to rewrite the expression: lim x → 0 4 1 [ f ( x ) ] 4 = 4 1 [ lim x → 0 f ( x ) ] 4 .
Substitute the given limit: 4 1 [ 4 ] 4 .
Simplify the expression: 4 1 × 256 = 64 .
The final answer is 64 .
Explanation
Problem Analysis We are given the limit of a function f ( x ) as x approaches 0, which is lim x → 0 f ( x ) = 4 . We need to find the limit of 4 1 [ f ( x ) ] 4 as x approaches 0. We will use limit properties to solve this problem.
Applying Limit Properties We can rewrite the expression using limit properties. The limit of a constant times a function is the constant times the limit of the function, and the limit of a function raised to a power is the limit of the function raised to that power. So, we have: x → 0 lim 4 1 [ f ( x ) ] 4 = 4 1 x → 0 lim [ f ( x ) ] 4
Moving Limit Inside Power Now, we can move the limit inside the power: 4 1 x → 0 lim [ f ( x ) ] 4 = 4 1 [ x → 0 lim f ( x ) ] 4
Substituting the Given Limit We are given that lim x → 0 f ( x ) = 4 . Substituting this into the expression, we get: 4 1 [ 4 ] 4
Simplifying the Expression Now, we simplify the expression: 4 1 [ 4 ] 4 = 4 1 [ 256 ] = 64
Final Answer Therefore, the limit of 4 1 [ f ( x ) ] 4 as x approaches 0 is 64.
Examples
In physics, consider a scenario where f ( x ) represents the intensity of a light source as a function of distance x from a detector. If lim x → 0 f ( x ) = 4 means the light intensity approaches 4 units as you get very close to the source, then lim x → 0 4 1 [ f ( x ) ] 4 calculates how the fourth power of that intensity, scaled by a factor of 4 1 , behaves near the source. This type of calculation is useful in understanding how certain physical quantities (like energy density, which can be proportional to the fourth power of intensity) change in extreme conditions, such as near a very bright light source.
