Determine the continuity of the functions F1 and F2 at the point (1, 2).
Understand the Problem
The question involves determining the continuity of two functions at a specific point and performing calculations related to these functions. It asks to check the condition for continuity and provide solutions for given mathematical expressions.
Answer
The answers are: 1. Continuous 2. Not continuous
Answer for screen readers
For the first case, (F_1) and (F_2) are continuous. For the second case, (F_1) and (F_2) are not continuous.
Steps to Solve
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Evaluate (F_1) at the Point (1,2)
Substitute (x = 1) and (y = 2) into the equation for (F_1): [ F_1 = \frac{x^2 + 2y}{x + y^2} = \frac{1^2 + 2(2)}{1 + 2^2} = \frac{1 + 4}{1 + 4} = \frac{5}{5} = 1 ]
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Check Continuity for (F_2 = I)
For (F_2 = 1), evaluate if (F_1) equals (F_2): [ F_1 = F_2 \Rightarrow 1 = 1 ] Hence, (F_1) and (F_2) are continuous.
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Evaluate (F_1) for the Second Function
Use the second function ( F_1 = 2x^2 + mx ) with ( y = mx ) and check continuity at (1, 2). Substitute: [ F_1 = 2(1)^2 + m(1) = 2 + m ]
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Evaluate (F_2 = 0)
Since (F_2 = 0), check consistency: [ F_1 \neq F_2 \Rightarrow 2 + m \neq 0 ]
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Conclusion on Continuity
Since (F_1) does not equal (F_2), we conclude: [ F_1 \neq 0 \Rightarrow \text{Not continuous} ]
For the first case, (F_1) and (F_2) are continuous. For the second case, (F_1) and (F_2) are not continuous.
More Information
Continuity requires that the limit of a function at a point matches the function's value at that point. The first function met this requirement while the second one did not.
Tips
- Forgetting Substitution: It is common to forget to fully substitute values into the functions. Ensure all variables are substituted correctly.
- Miscalculating Function Values: Double-check arithmetic calculations when substituting values to avoid mistakes in function evaluation.
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