Estimating Limit Values from Graphs - AP Calculus BC

1. Identify $\text{lim}_{x \to 1^+} f(x)$ from a graph approaching value 4.

   4. Right-hand limit reads $y$-value approached from positive side.

2. Estimate $\text{lim}_{x \to \text{infinity}} f(x)$ from a graph that levels off at $y = 9$.

   9. Horizontal asymptote at $y = 9$ as $x$ increases.

3. What does $\text{lim}_{x \to 0^-} f(x) = 3$ indicate about $f(x)$?

   $f(x)$ approaches 3 as $x$ approaches 0 from the left. Left-hand limit describes approach from negative direction.

4. Estimate $\text{lim}_{x \to 3^-} f(x)$ if the graph shows $f(x)$ approaching 6.

   6. Left-hand limit reads $y$-value from negative approach.

5. If $\text{lim}_{x \to c^-} f(x) \neq \text{lim}_{x \to c^+} f(x)$, what can be said about $\text{lim}_{x \to c} f(x)$?

   The limit does not exist. Two-sided limit requires both one-sided limits to be equal.

6. Estimate $\text{lim}_{x \to 4^-} f(x)$ from a graph showing $f(x)$ approaches 2.

   2. Left-hand limit reads $y$-value approached from negative side.

7. What is $\text{lim}_{x \to \text{infinity}} f(x)$ if the graph levels off at 7?

   7. Horizontal asymptotes show end behavior as $x \to \infty$.

8. What does the graph of $f(x)$ show if $\text{lim}_{x \to c} f(x) \neq f(c)$?

   A point discontinuity at $x = c$. When limit exists but doesn't equal function value at that point.

9. Estimate $\lim_{x \to -3^+} f(x)$ if the graph shows $f(x)$ approaching 5.

   5. Right-hand limit reads $y$-value from positive approach.

10. Estimate $\text{lim}_{x \to -3^+} f(x)$ if the graph shows $f(x)$ approaching 5.

    5. Right-hand limit reads $y$-value from positive approach.

11. What does it mean if $\text{lim}_{x \to c} f(x) = L$?

    As $x$ approaches $c$, $f(x)$ approaches $L$. Standard limit notation expressing function behavior near point $c$.

12. Estimate $\text{lim}_{x \to -\text{infinity}} f(x)$ if the graph approaches 0.

    0. Read where graph heads as $x$ decreases without bound.

13. Estimate $\text{lim}_{x \to \text{infinity}} f(x)$ from a graph that approaches $y = 0$.

    0. Horizontal asymptote shows end behavior approaching zero.

14. What is implied if $\text{lim}_{x \to \text{infinity}} f(x) = L$?

    The graph of $f(x)$ approaches $y = L$ as $x$ increases. Describes horizontal asymptote behavior at infinity.

15. Estimate $\lim_{x \to 5} f(x)$ if graph approaches different values from left and right.

    The limit does not exist. Different one-sided limits mean the two-sided limit doesn't exist.

16. What is $\text{lim}_{x \to 3} f(x)$ if the graph shows a hole at $x = 3$?

    The $y$-value the graph approaches as $x$ approaches 3. Holes don't affect limits; read where the curve would go.

17. What is indicated by $\text{lim}_{x \to c} f(x) = L$ on a continuous graph?

    The function value $f(c) = L$. Continuous functions have limits equal to function values.

18. What is the condition for a limit to exist at $x = c$?

    $\text{lim}_{x \to c^-} f(x) = \text{lim}_{x \to c^+} f(x)$. Both one-sided limits must exist and be equal.

19. Identify the limit of $f(x)$ as $x$ approaches 2 from the right on a graph.

    The $y$-value that $f(x)$ approaches as $x$ approaches 2 from the right. Read the $y$-coordinate where the curve heads as $x$ nears 2 from positive side.

20. Determine the limit if $\text{lim}_{x \to 0^-} f(x) = -4$ and $\text{lim}_{x \to 0^+} f(x) = 4$.

    The limit does not exist. Unequal one-sided limits mean no two-sided limit exists.

21. If a graph has a vertical asymptote at $x = 1$, what is $\text{lim}_{x \to 1} f(x)$?

    The limit does not exist. Vertical asymptotes mean function goes to $\pm\infty$, so limit doesn't exist.

22. Find $\text{lim}_{x \to 2} f(x)$ if the graph is continuous at $x = 2$ and $f(2) = 8$.

    8. For continuous functions, limit equals function value.

23. Estimate $\text{lim}_{x \to 0} f(x)$ from a graph with a vertical asymptote at $x = 0$.

    The limit does not exist. Vertical asymptotes cause limits to not exist.

24. Estimate $\text{lim}_{x \to 5} f(x)$ if graph approaches different values from left and right.

    The limit does not exist. Different one-sided limits mean the two-sided limit doesn't exist.

25. Which notation represents the limit of $f(x)$ as $x$ approaches 0 from the left?

    $\text{lim}_{x \to 0^-} f(x)$. The minus sign indicates approaching from the negative (left) side.

26. Determine $\lim_{x \to 0} f(x)$ if graph shows $f(x)$ approaches 3 from both sides.

    3. When both one-sided limits equal the same value.

27. Estimate $\text{lim}_{x \to -2} f(x)$ from the graph if $f(x)$ approaches 5.

    5. Read the $y$-value the graph approaches at $x = -2$.

28. Estimate $\text{lim}_{x \to 2} f(x)$ from a graph where $f(x)$ approaches $-3$.

    -3. Read the $y$-value the graph approaches at $x = 2$.

29. What is the implication if $\text{lim}_{x \to c} f(x)$ exists but $\text{lim}_{x \to c} f(x) \neq f(c)$?

    A removable discontinuity at $x = c$. Limit exists but function is not continuous at that point.

30. Estimate $\text{lim}_{x \to 7} f(x)$ if the graph shows a removable discontinuity at $x = 7$.

    The $y$-value the graph approaches, not the point at the discontinuity. Removable discontinuities don't affect limit values.

31. Estimate $\lim_{x \to 3^-} f(x)$ if the graph shows $f(x)$ approaching 6.

    6. Left-hand limit reads $y$-value from negative approach.

32. Estimate $\lim_{x \to \infty} f(x)$ from a graph that approaches $y = 0$.

    $0$. Horizontal asymptote shows end behavior approaching zero.

33. Determine $\text{lim}_{x \to 0} f(x)$ if graph shows $f(x)$ approaches 3 from both sides.

    3. When both one-sided limits equal the same value.

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