Estimating Limit Values from Graphs - AP Calculus AB

1. What is the limit of $f(x)$ as $x$ approaches a point of oscillation?

   The limit does not exist. Oscillating functions don't approach a single value.

2. Which condition ensures the existence of $\lim_{x \to c} f(x)$?

   Equal left and right limits at $c$. Both one-sided limits must match for existence.

3. What is $\lim_{x \to 0} \sin(\frac{1}{x})$?

   The limit does not exist due to oscillation. The function oscillates infinitely near $x = 0$.

4. Which term describes the behavior of $f(x)$ as $x \to 2^+$?

   Right-hand limit. The $+$ superscript indicates approach from the right side.

5. Identify $\lim_{x \to 0^-} \frac{1}{x}$.

   $-\infty$. Approaching from the left gives negative infinity.

6. Determine $\lim_{x \to 0} x^2$ using a graph.

   0. The function $x^2$ approaches $0$ as $x$ approaches $0$.

7. Determine $\lim_{x \to \infty} \frac{1}{x^2}$ using a graph.

   0. $\frac{1}{x^2}$ approaches $0$ as $x$ becomes large.

8. What is the limit of $f(x)$ as $x \to a$ for $f(x) = 3$?

   3. Constant functions have the same limit everywhere.

9. What is the limit of $\lim_{x \to -a} f(x)$ for $f(x) = x^2$?

   $a^2$. $(-a)^2 = a^2$ for any real number $a$.

10. What does $\lim_{x \to a} f(x) = f(a)$ imply?

    Function is continuous at $x = a$. The limit equals the function value at that point.

11. State the vertical asymptote implication for limits.

    Limit does not exist at the vertical asymptote. Vertical asymptotes prevent limit existence.

12. What is the limit as $x \to \infty$ of a linear function?

    Does not exist; increases or decreases without bound. Linear functions grow without bound at infinity.

13. What is indicated by $\lim_{x \to c} f(x) \neq f(c)$?

    A removable discontinuity at $x = c$. The limit exists but doesn't equal the function value.

14. How do you denote the limit of $f(x)$ as $x$ approaches $a$ from the right?

    $\lim_{x \to a^+} f(x)$. The plus superscript denotes right-hand limit notation.

15. State the meaning of $\lim_{x \to c} f(x) = L$.

    $f(x)$ approaches $L$ as $x$ approaches $c$. This is the formal definition of a limit.

16. Determine $\lim_{x \to 4} f(x)$ if $f(x)$ is continuous at $x = 4$.

    $f(4)$. Continuity means the limit equals the function value.

17. What does a continuous graph imply about limits at all points?

    Limits exist and equal the function values. Continuity guarantees limit existence everywhere.

18. State the notation for the left-hand limit of $f(x)$ at $x = a$.

    $\lim_{x \to a^-} f(x)$. Standard notation for left-hand limits.

19. What is the behavior of $f(x)$ near $x = a$ if $\lim_{x \to a} f(x) = L$?

    Approaches $L$ as $x$ approaches $a$. This describes the fundamental limit behavior.

20. What does $\lim_{x \to a} f(x)$ equal if $f(x)$ is linear?

    $f(a)$. Linear functions are continuous everywhere.

21. What does $\lim_{x \to c^-} f(x)$ represent?

    Limit from the left of $c$. The minus superscript indicates approach from the left.

22. What must be true for $\lim_{x \to c} f(x) = L$ to exist?

    Left and right limits at $c$ must be equal to $L$. Both one-sided limits must exist and be equal.

23. What does $\lim_{x \to 0} e^{-x}$ approach?

    1. $e^{-x}$ approaches $e^0 = 1$ as $x \to 0$.

24. What is the result of $\lim_{x \to 2^-} f(x)$ if $f(x)$ jumps at $x = 2$?

    The $y$-value approaching from the left of $x = 2$. One-sided limits can exist even with jump discontinuities.

25. What does $\lim_{x \to \infty} \frac{1}{x}$ equal?

    0. $\frac{1}{x}$ approaches $0$ as $x$ becomes large.

26. What is $\lim_{x \to c} f(x)$ if $f(x)$ is undefined at $x = c$?

    The $y$-value $f(x)$ approaches as $x$ nears $c$. Limits exist even when function values don't.

27. How do you denote the limit of $f(x)$ as $x$ approaches $a$ on a graph?

    $\lim_{x \to a} f(x)$. Standard mathematical notation for limits.

28. What is $\lim_{x \to 0} \frac{1}{x}$?

    Does not exist; approaches $\infty$ as $x \to 0^+$ and $-\infty$ as $x \to 0^-$. Left and right limits differ, so the limit doesn't exist.

29. How do you find $\lim_{x \to 5} f(x)$ if $f(x)$ is constant?

    The constant value. Constant functions have the same value everywhere.

30. How do you find $\lim_{x \to \infty} f(x)$ from a graph?

    Identify the horizontal asymptote if it exists. Look for the horizontal line the graph approaches.

31. How does $\lim_{x \to 0} \frac{1}{x^2}$ behave?

    Approaches $\infty$. $\frac{1}{x^2}$ grows without bound near $x = 0$.

32. Identify $\lim_{x \to -\infty} f(x)$ from the graph.

    The left horizontal asymptote value if it exists. Check where the graph levels off on the left side.

33. What is the limit of $f(x)$ as $x \to 3$ if $f(x)$ has a vertical asymptote at $x = 3$?

    The limit does not exist. Vertical asymptotes indicate unbounded behavior.

34. What is the graphical significance of $\lim_{x \to c} f(x) = \infty$?

    Vertical asymptote at $x = c$. Infinite limits create vertical asymptotes on graphs.

Study these flashcards