Find a and b so that the limit as x approaches zero of (a cos x + b x sin x - 5) / (x^4) exists, and also find the limit.
Understand the Problem
The question is asking to find the constants a and b such that the limit of the expression as x approaches zero exists. It also asks for the limit itself under the specified conditions.
Answer
The values of the constants are \( a = 0 \) and \( b = 0 \). The limit is \( 0 \).
Answer for screen readers
The values of the constants are ( a = 0 ) and ( b = 0 ). The limit is ( 0 ).
Steps to Solve
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Identify the expression and find the limit We need to analyze the expression given in the problem as ( x ) approaches zero. Let's denote the expression as ( f(x) = \frac{ax + b}{x^2} ). We want to ensure that this limit exists as ( x \to 0 ).
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Simplify the expression Before taking the limit, simplify the expression. If ( f(x) ) approaches a finite value, both the numerator and the denominator cannot go to zero simultaneously as ( x ) approaches zero.
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Set conditions for limit existence To ensure ( f(x) ) has a limit as ( x \to 0 ), the numerator must go to zero when the denominator goes to zero. Thus, we set ( ax + b = 0 ) when ( x = 0 ), which implies ( b = 0 ) to prevent an undefined form.
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Plug in the value of b and take the limit With ( b = 0 ), the function simplifies to ( f(x) = \frac{ax}{x^2} = \frac{a}{x} ). Now we can analyze ( \lim_{x \to 0} f(x) ).
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Determine the value of a For ( \lim_{x \to 0} \frac{a}{x} ) to exist and be a finite number, we need ( a = 0 ).
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Calculate the limit So, substituting ( a = 0 ) and ( b = 0 ) back into the limit gives us ( \lim_{x \to 0} f(x) = 0 ).
The values of the constants are ( a = 0 ) and ( b = 0 ). The limit is ( 0 ).
More Information
This type of limit problem is common in calculus, particularly when dealing with indeterminate forms. Understanding how to manipulate expressions to find their limits is a key skill in this subject. The solution shows how to tackle the behavior of functions near critical points such as ( x = 0 ).
Tips
- Assuming the limit exists without checking if ( ax + b ) makes the numerator zero when the denominator approaches zero.
- Failing to simplify the expression before taking the limit.
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