Determine the following calculus problems: 1. Determine $\lim_{x \to 4} f(x)$, where $f(x) = \begin{cases} 1, & \text{if } x \neq 4 \\ -1, & \text{if } x = 4 \end{cases}$ 2. For th... Determine the following calculus problems: 1. Determine $\lim_{x \to 4} f(x)$, where $f(x) = \begin{cases} 1, & \text{if } x \neq 4 \\ -1, & \text{if } x = 4 \end{cases}$ 2. For the function $f(x)$ in the graph below, determine the following: a. $\lim_{x \to 2^-} f(x)$ b. $\lim_{x \to 2^+} f(x)$ c. $\lim_{x \to -2^+} f(x)$ d. $f(2)$ 3. Use the graph to find the limit, if it exists. a. $\lim_{x \to ?} f(x)$ b. $\lim_{x \to ?} f(x)$ c. $\lim_{x \to ?} f(x)$
Understand the Problem
The image contains multiple calculus questions related to limits and function evaluation. Question 5 asks to determine the limit of a piecewise function. Question 6 involves finding limits from the left and right, and function evaluation from a graph. Question 7 is partially visible and asks to find limits from a graph.
Answer
d. $-8$ 5. $1$ 6. a. $2$ b. $-1$ c. $0$ d. $-1$
Answer for screen readers
d. $\lim_{x \to -2} (4 - 3x^2) = -8$
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$\lim_{x \to 4} f(x) = 1$
a. $\lim_{x \to 2^-} f(x) = 2$ b. $\lim_{x \to 2^+} f(x) = -1$ c. $\lim_{x \to -2^+} f(x) = 0$ d. $f(2) = -1$
Steps to Solve
- Evaluate limit in question d
To evaluate $\lim_{x \to -2} (4 - 3x^2)$, we substitute $x = -2$ into the expression $4 - 3x^2$.
$$ 4 - 3(-2)^2 = 4 - 3(4) = 4 - 12 = -8 $$
- Evaluate limit in question 5
To evaluate $\lim_{x \to 4} f(x)$ where $f(x) = \begin{cases} 1, & \text{if } x \neq 4 \ -1, & \text{if } x = 4 \end{cases}$, since we are evaluating the limit as $x$ approaches 4, we consider values of $x$ close to 4 but not equal to 4. Therefore, $f(x) = 1$.
$$ \lim_{x \to 4} f(x) = 1 $$
- Evaluate limits and function value in question 6
From the graph, we need to determine the following:
a. $\lim_{x \to 2^-} f(x)$: As $x$ approaches 2 from the left, $f(x)$ approaches 2. So, $\lim_{x \to 2^-} f(x) = 2$. b. $\lim_{x \to 2^+} f(x)$: As $x$ approaches 2 from the right, $f(x)$ approaches -1. So, $\lim_{x \to 2^+} f(x) = -1$. c. Since it seems like the question is asking for the limit as x approaches -2 from the right, or $\lim_{x \to -2^+} f(x)$: As $x$ approaches -2 from the right, $f(x)$ approaches 0. So, $\lim_{x \to -2^+} f(x) = 0$. d. $f(2)$: From the graph, we see that the value of the function at $x = 2$ is $f(2) = -1$.
d. $\lim_{x \to -2} (4 - 3x^2) = -8$
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$\lim_{x \to 4} f(x) = 1$
a. $\lim_{x \to 2^-} f(x) = 2$ b. $\lim_{x \to 2^+} f(x) = -1$ c. $\lim_{x \to -2^+} f(x) = 0$ d. $f(2) = -1$
More Information
Limits describe the value that a function approaches as the input approaches some value. For a limit to exist, the left-hand and right-hand limits must be equal. Function evaluation is simply finding the y-value of the function at a given x-value.
Tips
A common mistake is confusing the limit of a function as $x$ approaches a value with the value of the function at that point. In problem 5, $f(4) = -1$, but $\lim_{x \to 4} f(x) = 1$ because we consider values of x near 4 but not equal to 4 when evaluating the limit. Another common mistake with piecewise functions is not using the correct piece of the function when evaluating.
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