Math Problem Statement
a. Graph f(x) = { 4 - x^2, x ≠ -2; 7, x = -2 } b. Find lim f(x) as x approaches -2 from the left and right. c. Does lim f(x) exist as x approaches -2?
Solution
Let's break down the problem step by step.
Given Function:
\begin{cases} 4 - x^2, & \text{if } x \neq -2 \\ 7, & \text{if } x = -2 \end{cases}$$ --- ### Part (a) - Graph $$f(x)$$ The function is piecewise: - For $$x \neq -2$$, the function is $$f(x) = 4 - x^2$$, a parabola opening downward. - For $$x = -2$$, the function is $$f(x) = 7$$, which creates a single point at $$(-2, 7)$$. When graphing: - Plot the parabola $$f(x) = 4 - x^2$$, which is symmetric around the y-axis. - At $$x = -2$$, instead of following the parabola's value at $$x = -2$$, place a dot at $$(-2, 7)$$. --- ### Part (b) - Limits as $$x \to -2^-$$ and $$x \to -2^+$$ #### Left-hand limit ($$x \to -2^-$$): This is the limit of $$f(x) = 4 - x^2$$ as $$x$$ approaches $$-2$$ from the left. $$\lim_{x \to -2^-} f(x) = 4 - (-2)^2 = 4 - 4 = 0$$ #### Right-hand limit ($$x \to -2^+$$): This is the limit of $$f(x) = 4 - x^2$$ as $$x$$ approaches $$-2$$ from the right. $$\lim_{x \to -2^+} f(x) = 4 - (-2)^2 = 0$$ Thus, both the left-hand and right-hand limits are equal to 0. --- ### Part (c) - Does $$\lim_{x \to -2} f(x)$$ exist? The overall limit exists if both one-sided limits are equal. From part (b), we found: $$\lim_{x \to -2^-} f(x) = \lim_{x \to -2^+} f(x) = 0$$ However, $$f(x)$$ is defined as 7 when $$x = -2$$, which causes a discontinuity. Therefore, even though the limits from both sides are the same, the function's value at $$x = -2$$ differs from this limit. Thus, $$\lim_{x \to -2} f(x) = 0$$, but $$f(-2) = 7$$. Therefore, the limit exists, but the function is not continuous at $$x = -2$$. --- Would you like a more detailed graph, or do you have any other questions? ### Related Questions: 1. What is the difference between a limit and the actual value of a function at a point? 2. How do you determine continuity at a point for a piecewise function? 3. What happens when one-sided limits are not equal? 4. How do you graph piecewise functions with discontinuities? 5. Can a function have a limit but still be discontinuous at that point? ### Tip: When graphing a piecewise function, always pay attention to the point where the function changes its definition, as it often creates discontinuities or special points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Discontinuity
Formulas
Limit definition: lim x→a f(x)
f(x) = 4 - x^2 for x ≠ -2
f(x) = 7 for x = -2
Theorems
Limit existence condition: Left-hand limit = Right-hand limit
Continuity and Discontinuity at a point
Suitable Grade Level
High School - Calculus
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