1. Evaluate lim (x^2-5x+6)/(x^2-4) as x approaches 2. 2. If 5^2+x = 125, the value of x =? 3. Associativity property belongs to: A ∪ B = B ∪ A. 1. Find the slope of the line whose... 1. Evaluate lim (x^2-5x+6)/(x^2-4) as x approaches 2. 2. If 5^2+x = 125, the value of x =? 3. Associativity property belongs to: A ∪ B = B ∪ A. 1. Find the slope of the line whose inclination is π/4. 2. Evaluate lim (1/(x-1) + 2/(1-x^2)) as x approaches 1. 3. If U = {1,2,3,4,5,6,7,8,9}; A = {1,3,5,7}; B = {2,3,4,6,8}; C = {6,7,8,9}, find (A ∪ B)' and (A ∩ C)'.
Understand the Problem
The question consists of a math exam with various types of questions including limits, equations, geometry, and set theory. It requires solving or evaluating each problem presented. The key concepts include limits, properties of functions, domain and range, and set operations.
Answer
1. The answer to the limit question is $$ -\frac{1}{2} $$. 2. The value of \( x \) is $$ x = \frac{\log(118)}{\log(5)} $$.
Answer for screen readers
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The answer to the limit question is $$ -\frac{1}{2} $$.
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The value of ( x ) for ( 52^x + 7 = 125 ) is $$ x = \frac{\log(118)}{\log(5)} $$.
Steps to Solve
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Evaluate the limit We start with the limit given in the first question: $$ \lim_{x \to 2} \frac{x^2 - 5x + 6}{2x - 4} $$ We can simplify this by factoring the numerator.
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Factor the numerator The numerator factors as: $$ x^2 - 5x + 6 = (x - 2)(x - 3) $$ So the expression becomes: $$ \lim_{x \to 2} \frac{(x - 2)(x - 3)}{2(x - 2)} $$
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Canceling terms We can cancel the common factor $(x - 2)$ from the numerator and denominator (this is valid since we are taking the limit as $x$ approaches 2 but not equal to 2): $$ \lim_{x \to 2} \frac{x - 3}{2} $$
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Substituting the limit Now substitute $x = 2$ in the simplified expression: $$ \frac{2 - 3}{2} = \frac{-1}{2} $$
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Final result Hence, the answer to the first part is: $$ -\frac{1}{2} $$
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Solving for x in the equation The second question involves finding the value of $x$ from the equation $52^x + 7 = 125$. Subtract 7 from both sides: $$ 52^x = 118 $$
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Taking logarithms To solve for $x$, take the logarithm of both sides: $$ x \log(5) = \log(118) $$
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Divide by log(5) Finally, divide by $\log(5)$: $$ x = \frac{\log(118)}{\log(5)} $$
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The answer to the limit question is $$ -\frac{1}{2} $$.
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The value of ( x ) for ( 52^x + 7 = 125 ) is $$ x = \frac{\log(118)}{\log(5)} $$.
More Information
The limit problem demonstrates the technique of factoring to simplify expressions before taking limits. The second question highlights the use of logarithms to solve exponential equations, which is a fundamental concept in algebra.
Tips
- Forgetting to simplify the limit before substituting can lead to errors in the final answer.
- Confusing the properties of logarithms, which can result in incorrect evaluations or manipulations.
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