Exemplo 2 de Precalculus, 12.1 Finding Limits: Numerical and Graphical Approaches

Finding a Limit Using a Graph

Problema e solução em inglês, como no livro original.

Determine the following limits and function value for the function f shown in Figure 6.
1. $\lim_{x \to 2^{-}} f (x)$
2. $\lim_{x \to 2^{+}} f (x)$
3. $\lim_{x \to 2} f (x)$
4. $f (2)$
Figure 6
Determine the following limits and function value for the function f shown in Figure 7.
1. $\lim_{x \to 2^{-}} f (x)$
2. $\lim_{x \to 2^{+}} f (x)$
3. $\lim_{x \to 2} f (x)$
4. $f (2)$
Figure 7

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

1
Solução
1. Looking at Figure 6:
  1. $\lim_{x \to 2^{-}} f (x) = 8;$ when $x < 2,$ but infinitesimally close to 2, the output values get close to $y = 8.$
  2. $\lim_{x \to 2^{+}} f (x) = 3;$ when $x > 2,$ but infinitesimally close to 2, the output values approach $y = 3.$
  3. $\lim_{x \to 2} f (x)$ does not exist because $\lim_{x \to 2^{-}} f (x) \neq \lim_{x \to 2^{+}} f (x);$ the left and right-hand limits are not equal.
  4. $f (2) = 3$ because the graph of the function $f$ passes through the point $(2, f (2))$ or $(2, 3) .$
2. Looking at Figure 7:
  1. $\lim_{x \to 2^{-}} f (x) = 8;$ when $x < 2$ but infinitesimally close to 2, the output values approach $y = 8.$
  2. $\lim_{x \to 2^{+}} f (x) = 8;$ when $x > 2$ but infinitesimally close to 2, the output values approach $y = 8.$
  3. $\lim_{x \to 2} f (x) = 8$ because $\lim_{x \to 2^{-}} f (x) = \lim_{x \to 2^{+}} f (x) = 8;$ the left and right-hand limits are equal.
  4. $f (2) = 4$ because the graph of the function $f$ passes through the point $(2, f (2))$ or $(2, 4) .$

Sua vez

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Using the graph of the function $y = f (x)$ shown in Figure 8, estimate the following limits.

$lim_{x \to 0^{-}} f (x)$
$lim_{x \to 0^{+}} f (x)$
$\underset{x \to 0}{l i m} f (x)$
$lim_{x \to 2^{-}} f (x)$
$lim_{x \to 2^{+}} f (x)$
$\underset{x \to 2}{l i m} f (x)$
$lim_{x \to 4^{-}} f (x)$
$lim_{x \to 4^{+}} f (x)$
$\underset{x \to 4}{l i m} f (x)$

Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)

Figure 8

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