Math Problem Statement
If possible, choose k so that the following function is continuous on any interval: f(x) = (6x^5 - 30x^4)/(x - 5) when x ≠ 5, and k when x = 5.
Solution
To ensure that the given piecewise function is continuous on any interval, the value of must be chosen such that the function is continuous at . This means that the left-hand limit (as ) of the first part of the function must equal the value of the function at , which is .
The function is given as:
\begin{cases} \frac{6x^5 - 30x^4}{x - 5} & x \neq 5 \\ k & x = 5. \end{cases}$$ ### Steps: 1. First, simplify the expression $$\frac{6x^5 - 30x^4}{x - 5}$$ for $$x \neq 5$$. 2. Then, compute the limit of this simplified expression as $$x \to 5$$. 3. Set that limit equal to $$k$$ and solve for $$k$$. Let's simplify the expression first. The numerator is $$6x^5 - 30x^4$$, which can be factored. $$6x^5 - 30x^4 = 6x^4(x - 5)$$ Now, substituting back into the function: $$\frac{6x^5 - 30x^4}{x - 5} = \frac{6x^4(x - 5)}{x - 5}$$ For $$x \neq 5$$, we can cancel the $$(x - 5)$$ term: $$\frac{6x^4(x - 5)}{x - 5} = 6x^4$$ Thus, the simplified form of the function for $$x \neq 5$$ is: $$f(x) = 6x^4 \quad \text{for} \quad x \neq 5.$$ ### Step 2: Calculate the limit as $$x \to 5$$. $$\lim_{x \to 5} 6x^4 = 6(5)^4 = 6 \times 625 = 3750.$$ ### Step 3: Set the limit equal to $$k$$ for continuity at $$x = 5$$. $$k = 3750.$$ Thus, to make the function continuous at $$x = 5$$, the value of $$k$$ must be: $$k = 3750.$$ ### Would you like further details or clarifications? Here are 5 related questions for practice: 1. What is the definition of a continuous function? 2. Why is factoring important when simplifying rational functions? 3. How do you determine the limit of a polynomial as $$x \to a$$? 4. What is the importance of left-hand and right-hand limits in continuity? 5. How would you handle discontinuities caused by other types of functions? **Tip**: Always check the limit from both sides when evaluating continuity at a specific point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Continuity
Limits
Formulas
Limit of a function as x approaches a value
Factoring polynomials
Continuity condition: limit of f(x) as x → c = f(c)
Theorems
Definition of continuity
Factor theorem
Suitable Grade Level
Grades 11-12 (Pre-Calculus, Calculus)