Continuity Analysis of a Piecewise Function with Trigonometric Terms

Math Problem Statement

Studiaza continuitatea functiei

The function shown in the image appears to be defined as follows:

$f: \mathbb{R} \to \mathbb{R}$

\frac{\tan(x)}{x}, & \text{if } x \neq 0 \\ 2, & \text{if } x = 0 \end{cases}$$ You are asked to study the continuity of this function. ### Continuity at $$x = 0$$: To check for continuity at $$x = 0$$, we need to verify if the following condition holds: $$\lim_{x \to 0} f(x) = f(0)$$ From the definition, $$f(0) = 2$$. Now, let's check the limit as $$x \to 0$$. - For $$x \neq 0$$, $$f(x) = \frac{\tan(x)}{x}$$. We know from the standard limit: $$\lim_{x \to 0} \frac{\tan(x)}{x} = 1$$ Thus: $$\lim_{x \to 0} f(x) = 1$$ ### Conclusion: Since: $$\lim_{x \to 0} f(x) = 1 \quad \text{and} \quad f(0) = 2$$ the function is **not continuous** at $$x = 0$$, because the limit does not match the value of the function at $$x = 0$$. Would you like a more detailed explanation or any further steps in the analysis of the function? Here are a few related questions that might interest you: 1. What is the general condition for continuity at a point for any function? 2. How would the function behave near $$x = 0$$ if it were continuous? 3. Can the function be made continuous at $$x = 0$$? If so, how? 4. What is the behavior of the function at $$x$$ approaching other points like $$\pi$$, $$-\pi$$, etc.? 5. What role does the tangent function play in the continuity of this function? Tip: For functions with piecewise definitions, always check both the limit and the function value at the point where the pieces meet.

Drop file here or Click Here to upload

Continuity
Limits
Piecewise Functions

lim_{x \to c} f(x) = f(c)
lim_{x \to 0} \frac{\tan(x)}{x} = 1

Definition of Continuity
Limit of Trigonometric Functions

Grades 11-12

Study the Continuity of the Piecewise Function f(x) = tan(x)/x for x ≠ 0 and f(0) = 2

Analyzing Continuity and Tangent Lines for Piecewise Function f(x)

Understanding Piecewise Function with 1/x and Continuity at x = 0

Continuity of a Piecewise Function: sin(x)/x at x = 0

Continuity Analysis of a Piecewise Function