Determine [tex]\lim _{z \rightarrow 5} \frac{\sqrt{x^2-9}}{8 x}[/tex]
a) [tex]\frac{1}{10}[/tex]
b) 4
c) 10
Answer (2)
Check if the function is continuous at x = 5 .
Substitute x = 5 into the function 8 x x 2 − 9 .
Simplify the expression 8 ( 5 ) 5 2 − 9 = 40 16 = 40 4 .
The limit of the function as x approaches 5 is 10 1 .
Explanation
Problem Analysis and Continuity We are asked to find the limit of the function f ( x ) = 8 x x 2 − 9 as x approaches 5. First, we need to check if the function is continuous at x = 5 .
Checking for Continuity Since the function involves a square root, we need to ensure that the expression inside the square root is non-negative. In this case, x 2 − 9 should be greater than or equal to 0. When x = 5 , x 2 − 9 = 5 2 − 9 = 25 − 9 = 16 , which is positive. Also, the denominator 8 x should not be zero. When x = 5 , 8 x = 8 ( 5 ) = 40 , which is not zero. Therefore, the function is continuous at x = 5 .
Direct Substitution Since the function is continuous at x = 5 , we can find the limit by direct substitution. We substitute x = 5 into the function:
x → 5 lim 8 x x 2 − 9 = 8 ( 5 ) 5 2 − 9
Simplifying the Expression Now, we simplify the expression:
8 ( 5 ) 5 2 − 9 = 40 25 − 9 = 40 16 = 40 4 = 10 1
Final Answer Therefore, the limit of the function as x approaches 5 is 10 1 .
Examples
Imagine you are designing a bridge and need to calculate the stress on a support beam as the load approaches a certain value. The function representing the stress might be similar to the one in this problem, and finding the limit would tell you the maximum stress the beam will experience, ensuring the bridge's safety.
The limit of 8 x x 2 − 9 as x approaches 5 is 10 1 . This was determined by checking for continuity and using direct substitution. Hence, the correct answer is option a) 10 1 .
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