Math Problem Statement
The user uploaded an image with calculus questions related to continuity, limits, and function analysis.
Solution
I see you've uploaded a calculus worksheet in Indonesian. Here’s a breakdown of each question, with explanations to help solve them.
1. Problem:
Given:
-1, & \text{if } x \leq 0 \\ ax + b, & \text{if } 0 < x < 1 \\ 1, & \text{if } x \geq 1 \end{cases}$$ Determine values of $$a$$ and $$b$$ so that $$f(x)$$ is continuous at $$x = 0$$ and $$x = 1$$. **Solution:** To ensure continuity, $$\lim_{x \to 0^+} f(x) = f(0)$$ and $$\lim_{x \to 1^-} f(x) = f(1)$$. Set up equations for each condition, then solve for $$a$$ and $$b$$. --- ### 2. **Problem:** Given: $$f(x) = \frac{x + 2}{x^2 - 4}$$ Determine the points where $$f(x)$$ is discontinuous. **Solution:** Factor $$x^2 - 4 = (x - 2)(x + 2)$$ in the denominator to find points where $$f(x)$$ is undefined, as these are likely discontinuities. --- ### 3. **Problem:** Given: $$f(x) = \begin{cases} \frac{x^3 - 27}{x - 3}, & x \neq 3 \\ 23, & x = 3 \end{cases}$$ Determine if $$f(x)$$ is continuous at $$x = 3$$. **Solution:** Find $$\lim_{x \to 3} f(x)$$ by simplifying $$\frac{x^3 - 27}{x - 3}$$ using factorization. Compare it with $$f(3) = 23$$. --- ### 4. **Problem:** Calculate: $$\lim_{x \to 0} \sqrt{x} \left( 1 + \sin^2 \left( \frac{1}{x} \right) \right)$$ **Solution:** Analyze the behavior of $$\sin^2 \left( \frac{1}{x} \right)$$ as $$x \to 0$$ and use properties of limits to evaluate the expression. --- ### 5. **Problem:** Given: $$f(x) = \frac{x + 3}{|x^2 + 3x|}$$ Find the points where $$f(x)$$ is discontinuous. **Solution:** Analyze where $$x^2 + 3x = 0$$ (points where the denominator is zero), which are potential discontinuities. --- ### 6. **Problem:** Use theorems to prove that the function: $$H(x) = \cos(e^{\sqrt{x}})$$ is continuous in its domain. **Solution:** Demonstrate continuity by proving that the composition of continuous functions is continuous. --- Would you like a detailed solution for any specific problem or have any questions? ### Additional Practice Questions: 1. How do you determine if a piecewise function is continuous at boundary points? 2. What steps can be taken to find discontinuities in rational functions? 3. How can limits help determine continuity in functions involving absolute values? 4. What techniques can simplify complex limits involving trigonometric functions? 5. How do compositions of functions affect continuity? ### Tip: For continuity at a point, ensure that the left-hand limit, right-hand limit, and the function’s value at the point are all equal.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Discontinuities
Composition of Functions
Formulas
Continuity condition: lim(x→c^-) f(x) = f(c) = lim(x→c^+) f(x)
Limit calculations with trigonometric functions
Simplifying rational functions for continuity analysis
Composition of continuous functions
Theorems
Continuity of Piecewise Functions
Continuity of Composite Functions
Suitable Grade Level
Undergraduate (Calculus I)