What is $\lim _{x \rightarrow 4}(3 x-3)^{\frac{5}{2}} ?$
Answer (2)
The limit of the expression ( 3 x − 3 ) 2 5 as x approaches 4 is 243. This result is obtained by substituting x = 4 into the expression and simplifying it. The final calculation shows that 9 2 5 = 243 .
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Substitute x = 4 into the function ( 3 x − 3 ) 2 5 .
Simplify the expression: ( 3 ( 4 ) − 3 ) 2 5 = ( 9 ) 2 5 .
Calculate 9 2 5 = ( 3 ) 5 = 243 .
The limit of the function as x approaches 4 is 243 .
Explanation
Problem Analysis We are asked to find the limit of the function ( 3 x − 3 ) 2 5 as x approaches 4. Since this is a polynomial function, we can directly substitute the value of x into the function to find the limit.
Substitution Substitute x = 4 into the expression ( 3 x − 3 ) 2 5 : ( 3 ( 4 ) − 3 ) 2 5 = ( 12 − 3 ) 2 5 = ( 9 ) 2 5
Simplification Now, we simplify the expression: 9 2 5 = ( 9 2 1 ) 5 = ( 3 ) 5 = 3 × 3 × 3 × 3 × 3 = 243
Final Answer Therefore, the limit of the function as x approaches 4 is 243.
Examples
In physics, when analyzing the motion of an object, you might encounter functions that describe its position or velocity. Evaluating the limit of such a function as time approaches a certain value helps predict the object's state at that specific moment. For example, if the velocity function is v ( t ) = ( 3 t − 3 ) 2 5 , finding lim t → 4 v ( t ) tells us the velocity of the object at t = 4 . This is crucial for understanding the object's behavior and predicting its future trajectory.
