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Introduction to Limits quiz

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  • What does the notation lim_{x→c} f(x) represent?
    It represents the limit of the function f(x) as x approaches the value c, or the y-value f(x) approaches as x gets close to c.
  • How can you estimate a limit using a graph?
    You look at the y-value the function approaches as x gets close to the target value from both sides on the graph.
  • What is a one-sided limit?
    A one-sided limit examines the behavior of a function as x approaches a value from only one side, either the left or the right.
  • How is the left-sided limit of f(x) as x approaches c written in notation?
    It is written as lim_{x→c^-} f(x), indicating x approaches c from the left.
  • How is the right-sided limit of f(x) as x approaches c written in notation?
    It is written as lim_{x→c^+} f(x), indicating x approaches c from the right.
  • When does the limit of f(x) as x approaches c exist?
    The limit exists if the function approaches the same y-value from both the left and right as x approaches c.
  • What does it mean if the left and right limits as x approaches c are not equal?
    It means the overall limit as x approaches c does not exist (DNE).
  • Why can't you always find a limit by plugging in the value of c into the function?
    Because the function value at c may not match the value the function approaches as x gets close to c, or the function may not even be defined at c.
  • What is a common scenario where a limit does not exist due to a 'jump'?
    A limit does not exist at a point where a piecewise function has a jump, meaning the left and right limits are different.
  • What happens to the limit if the function approaches infinity as x approaches c?
    The limit does not exist because infinity is not a number; this is called unbounded behavior.
  • What type of function behavior near c causes the limit to not exist due to oscillation?
    If the function oscillates (goes up and down repeatedly) as x approaches c, the limit does not exist.
  • How can you use a table of values to estimate a limit?
    By plugging in values of x that get closer and closer to c from both sides and observing the y-values they approach.
  • If lim_{x→1^-} f(x) = -1 and lim_{x→1^+} f(x) = -1, what is lim_{x→1} f(x)?
    lim_{x→1} f(x) = -1, because the left and right limits are equal.
  • If lim_{x→3^-} f(x) = 1 and lim_{x→3^+} f(x) = 4, what can you say about lim_{x→3} f(x)?
    The limit does not exist because the left and right limits are not equal.
  • What are three common cases where a limit does not exist?
    Limits do not exist when there is a jump (piecewise function), unbounded behavior (approaching infinity), or oscillation near the point.