Estimating Limit Values from Graphs - AP Calculus AB

1. Based on the description of the graph of $f(x)$, what is the value of $\lim_{x \to 2} f(x)$?
   Answer: $5$ For a continuous function, the limit at a point is equal to the function's value at that point. Since the graph of $f(x)$ is continuous and passes through $(2, 5)$, we have $f(2) = 5$. Therefore, $\lim_{x \to 2} f(x) = f(2) = 5$.
2. Based on the description of the graph of $f(x)$, what is the value of $\lim_{x \to -2^-} f(x)$?
   Answer: $6$ The notation $x \to -2^-$ indicates the limit as $x$ approaches -2 from the left. The description states that as $x$ approaches -2 from the left, the $y$-values approach 6. Therefore, $\lim_{x \to -2^-} f(x) = 6$.
3. Based on the description of the graph of $f(x)$, what is the value of $\lim_{x \to 0} f(x)$?
   Answer: The limit does not exist. The two-sided limit $\lim_{x \to 0} f(x)$ does not exist because the function is not defined for $x < 0$. Therefore, the left-hand limit $\lim_{x \to 0^-} f(x)$ does not exist. For the two-sided limit to exist, both one-sided limits must exist and be equal. Note that the one-sided limit from the right, $\lim_{x \to 0^+} f(x)$, does exist and is equal to 2.
4. Based on the description of the graph of $g(x)$, what is $\lim_{x \to 2} g(x)$?
   Answer: $4$ The problem states that the graph of $g(x)$ is continuous for all real numbers. For a continuous function, the limit at any point is equal to the function's value at that point. Since the graph has a local maximum at $(2,4)$, the point $(2,4)$ is on the graph, meaning $g(2)=4$. Therefore, $\lim_{x \to 2} g(x) = g(2) = 4$.
5. Based on the description of the graph of $g(x)$, what is the value of $\lim_{x \to 3} g(x)$?
   Answer: $4$ The limit of a function as $x$ approaches a value is the $y$-value that the function approaches. The description states that as $x$ approaches 3, the $y$-values approach 4. The actual value of the function at $x=3$, which is $g(3)=1$, does not affect the limit.
6. Based on the descriptions of the graphs, what is the value of $\lim_{x \to 2} f(g(x))$?
   Answer: $5$ To evaluate the limit of a composite function, we first find the limit of the inner function: $\lim_{x \to 2} g(x) = -1$. Let $u = g(x)$; as $x \to 2$, $u \to -1$. The problem becomes finding $\lim_{u \to -1} f(u)$. Since $f(x)$ is continuous at $x=-1$, this limit is $f(-1)$, which is 5.
7. Based on the descriptions of the graphs, what is $\lim_{x \to -2} (f(x) - g(x))$?
   Answer: $3$ Using the properties of limits, $\lim_{x \to -2} (f(x) - g(x)) = \lim_{x \to -2} f(x) - \lim_{x \to -2} g(x)$. From the descriptions, $\lim_{x \to -2} f(x) = 4$ and $\lim_{x \to -2} g(x) = 1$. Therefore, the limit is $4 - 1 = 3$.
8. Based on the description of the graph of $h(x)$, what is the value of $\lim_{x \to -\infty} h(x)$?
   Answer: $-2$ The limit as $x \to -\infty$ describes the behavior of the function as $x$ moves to the far left. The description states that the graph approaches the line $y=-2$ as $x$ grows infinitely large in the negative direction. Therefore, $\lim_{x \to -\infty} h(x) = -2$.
9. Based on the description of the graph of $f(x)$, what is $\lim_{x \to 3} f(x)$?
   Answer: $\infty$ The statement that the graph increases without bound as $x$ approaches 3 means that the function values are approaching positive infinity. Since this is the behavior from both sides, the limit is $\infty$. While an infinite limit means the limit does not exist as a finite number, $\infty$ is the most precise description of the behavior.
10. From the graphical behavior described, what is the value of $\lim_{x \to \infty} f(x)$?
    Answer: $4$ A horizontal asymptote of a graph of a function describes the end behavior of the function. The statement that $y=4$ is a horizontal asymptote as $x \to \infty$ means that the function values $f(x)$ approach 4 as $x$ increases without bound. This is the definition of $\lim_{x \to \infty} f(x) = 4$.
11. Based on the description of the graph of $h(x)$, what is the value of $\lim_{x \to 1^+} h(x)$?
    Answer: $-\infty$ The question asks for the right-hand limit as $x$ approaches 1. The description states that the graph has a vertical asymptote at $x=1$ and that as $x$ approaches 1 from the right, the function decreases without bound. This means the function values approach negative infinity. The information about the removable discontinuity at $x=3$ is extraneous.
12. Based on the descriptions of the graphs, what is $\lim_{x \to 5} \frac{f(x)}{g(x)}$?
    Answer: $4$ Using the properties of limits, $\lim_{x \to 5} \frac{f(x)}{g(x)} = \frac{\lim_{x \to 5} f(x)}{\lim_{x \to 5} g(x)}$. From the descriptions, $\lim_{x \to 5} f(x) = 8$ and $\lim_{x \to 5} g(x) = 2$. Therefore, the limit of the quotient is $\frac{8}{2} = 4$.
13. Based on the description of the graph of $g(x)$, what is the value of $\lim_{x \to 4^+} g(x)$?
    Answer: $-3$ The notation $x \to 4^+$ indicates the limit as $x$ approaches 4 from values greater than 4 (from the right). For $x > 4$, the graph of $g(x)$ is the horizontal line $y=-3$. Therefore, as $x$ approaches 4 from the right, the function values approach -3.
14. Which of the following must be true about the graph of $f(x)$?
    Answer: $\lim_{x \to d} f(x)$ does not exist. At a jump discontinuity, the left-hand and right-hand limits are not equal. Since $\lim_{x \to d^-} f(x) = M$ and $\lim_{x \to d^+} f(x) = N$ with $M \neq N$, the two-sided limit $\lim_{x \to d} f(x)$ does not exist. For a removable discontinuity at $x=c$, the limit $L$ exists, but it is not necessarily equal to $f(c)$. The value of $f(d)$ could be anything and is not restricted to be $M$ or $N$.
15. Based on the descriptions of the graphs, what is $\lim_{x \to \infty} (f(x)g(x))$?
    Answer: $-3$ Using the properties of limits, $\lim_{x \to \infty} (f(x)g(x)) = (\lim_{x \to \infty} f(x))(\lim_{x \to \infty} g(x))$. From the descriptions, $\lim_{x \to \infty} f(x) = 3$ and $\lim_{x \to \infty} g(x) = -1$. Therefore, the limit of the product is $(3)(-1) = -3$.
16. Based on the description of the graph of $f(x)$, what is the value of $\lim_{x \to 0} f(x)$?
    Answer: $3$ The limit of a function at a point depends on the value the function approaches from both sides, not the actual value at the point. Since both the left-hand limit ($\lim_{x \to 0^-} f(x) = 3$) and the right-hand limit ($\lim_{x \to 0^+} f(x) = 3$) are equal, the two-sided limit exists and equals 3. This describes a removable discontinuity because the limit exists but differs from the function value at that point.

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