Finding Limits GraphicallyLet f(x) = {3 - x , x 2c. Find limx→−1− f(x) and limx→−1+ f(x).

Ch. 2 - Limits and Continuity

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Hass 15th Edition

Ch. 2 - Limits and Continuity

문제 2.4.4c

2장, 문제 2.4.4c

Finding Limits Graphically

Let f(x) = {3 - x , x < 2
2, x = 2
x/2, x > 2

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c. Find limx→−1− f(x) and limx→−1+ f(x).

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Identify the piecewise function f(x) and the conditions for each piece. Here, f(x) = 3 - x for x < 2 and f(x) = x/2 for x > 2.

Understand that the limit as x approaches -1 from the left, denoted as lim_{x→−1−} f(x), involves considering values of x that are slightly less than -1.

Since -1 is less than 2, use the piece of the function f(x) = 3 - x to evaluate the left-hand limit. Substitute x = -1 into this expression to find the limit.

For the limit as x approaches -1 from the right, denoted as lim_{x→−1+} f(x), consider values of x that are slightly greater than -1.

Again, since -1 is less than 2, use the same piece of the function f(x) = 3 - x to evaluate the right-hand limit. Substitute x = -1 into this expression to find the limit.

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity. Limits can be approached from the left (denoted as lim x→c−) or from the right (denoted as lim x→c+), which is crucial for analyzing piecewise functions.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In the given function f(x), different formulas apply for x values less than, equal to, or greater than 2. Understanding how to evaluate piecewise functions is essential for finding limits, especially when the limit point lies at a boundary between pieces.

Graphical Interpretation of Limits

Graphically interpreting limits involves analyzing the behavior of a function's graph as it approaches a specific x-value. This includes observing the values the function approaches from both the left and right sides. For the question at hand, examining the graph of f(x) will help determine the left-hand limit (lim x→−1− f(x)) and the right-hand limit (lim x→−1+ f(x)).

Finding Limits GraphicallyLet f(x) = {3 - x , x < 22, x = 2x/2, x > 2<IMAGE>c. Find limx→−1− f(x) and limx→−1+ f(x).