Math Problem Statement
Solution
I can see a collection of exercises related to continuity and limits of functions in the image you uploaded. Let's break down the first exercise and solve it:
Exercice 1:
We are given the function defined as:
\begin{cases} 3 \quad & \text{if} \quad x = 4 \\ \frac{x\sqrt{x} - 8}{x - 4} \quad & \text{if} \quad x \neq 4 \end{cases}$$ And we are asked to study the continuity of $$f$$ at the point $$a = 4$$. #### Step 1: Check Continuity at $$x = 4$$ A function is continuous at $$x = 4$$ if the following conditions are met: 1. $$f(4)$$ is defined. 2. The limit $$\lim_{x \to 4} f(x)$$ exists. 3. $$\lim_{x \to 4} f(x) = f(4)$$. #### Condition 1: Is $$f(4)$$ defined? From the function, $$f(4) = 3$$. So, $$f(4)$$ is indeed defined. #### Condition 2: Does $$\lim_{x \to 4} f(x)$$ exist? We need to compute the limit of the piecewise function as $$x \to 4$$ for the second case: $$\lim_{x \to 4} \frac{x\sqrt{x} - 8}{x - 4}$$ We can try simplifying this expression by substitution or factorization techniques. Would you like me to proceed with the detailed calculation of this limit? After this, I can move on to the next exercises or provide explanations for other parts. Let me know how you'd like to continue! --- Here are 5 related questions to expand your understanding: 1. What are the conditions for continuity at a point? 2. How can we simplify limits involving square roots? 3. What is the difference between left-hand and right-hand limits? 4. How does the definition of a piecewise function affect its continuity? 5. What is the role of indeterminate forms in limit calculations? **Tip:** When dealing with limits involving indeterminate forms, always try factoring, simplifying, or applying L'Hopital's rule if necessary.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Functions
Square Roots
Algebraic Simplification
Left-hand and Right-hand Limits
Formulas
Limit definition: lim_{x -> a} f(x)
Continuity condition: f(a) = lim_{x -> a} f(x)
Piecewise function evaluation
Simplification of expressions with square roots
Theorems
Continuity at a point
Indeterminate forms
Limit Laws
Suitable Grade Level
Grades 10-12
Related Recommendation
Exercises on Continuity, Limits, and Function Analysis
Limits, Continuity, and Differentiation of Functions – Math Exercises
Calculus Practice: Solving Limits and Checking Continuity
Understanding Limits and Continuity of Piecewise Functions
Tutorial on Limits and Continuity: Solved Problems and Key Concepts