Skip to main content
Pearson+ LogoPearson+ Logo

Introduction to Limits quiz #1 Flashcards

Introduction to Limits quiz #1
Control buttons has been changed to "navigation" mode.
1/16
  • What is the definition of a limit in calculus?
    A limit describes the value that a function approaches as the input (x) gets arbitrarily close to a specific value.
  • How can you determine the limit of a function as x approaches a value using a table of values?
    By plugging in values of x that get closer and closer to the target value from both sides and observing the trend in the corresponding y-values.
  • What is the difference between a one-sided limit and a regular (two-sided) limit?
    A one-sided limit considers the function's behavior as x approaches a value from only one side (left or right), while a regular limit requires the function to approach the same value from both sides.
  • What does the notation limₓ→c⁻ f(x) represent?
    It represents the left-sided limit of f(x) as x approaches c from values less than c.
  • What does the notation limₓ→c⁺ f(x) represent?
    It represents the right-sided limit of f(x) as x approaches c from values greater than c.
  • If the left-sided and right-sided limits as x approaches c are not equal, what can you conclude about the limit at c?
    The overall limit as x approaches c does not exist.
  • Why might the limit of a function at a point not exist?
    A limit might not exist if the function approaches different values from the left and right, has unbounded behavior (goes to infinity), or oscillates near the point.
  • How can a graph help you determine the limit of a function as x approaches a value?
    By visually observing the y-value the function approaches as x gets close to the target value from both sides.
  • Does the value of the function at x = c affect the limit as x approaches c?
    No, the limit depends only on the values as x gets close to c, not the value at c itself.
  • What is an example of a function where the limit as x approaches c does not exist due to a jump discontinuity?
    A piecewise function where the left and right limits at c are different, such as f(x) = x-2 for x<3 and f(x)=4 for x≥3 at x=3.
  • What is an example of a function where the limit as x approaches c does not exist due to unbounded behavior?
    A rational function with a vertical asymptote at x=c, such as f(x) = 1/(x-2) as x approaches 2.
  • What is an example of a function where the limit as x approaches c does not exist due to oscillation?
    Functions like sin(1/x) or cos(1/x) as x approaches 0, which oscillate infinitely near the point.
  • If limₓ→1⁻ f(x) = -1 and limₓ→1⁺ f(x) = -1, what is limₓ→1 f(x)?
    limₓ→1 f(x) = -1, since both one-sided limits are equal.
  • If limₓ→3⁻ f(x) = 1 and limₓ→3⁺ f(x) = 4, what is limₓ→3 f(x)?
    The limit does not exist because the left and right limits are not equal.
  • Can you always find the limit by plugging the value of x directly into the function?
    No, this only works if the function is continuous at that point; otherwise, you must analyze the behavior as x approaches the value.
  • Why is understanding limits important in calculus?
    Limits are fundamental for analyzing function behavior near specific points and are essential for defining derivatives and integrals.