Calc BC 1.1 Video

• What is the fundamental idea that calculus is based upon?


  The idea of a limit
• Why is the idea of a limit considered simple despite its importance?


  Because it can be easily understood and applied
• Define the function \( f(x) \) as given in the video.


  \( f(x) = \frac{x - 1}{x - 1} \)
• What happens when you try to simplify \( f(x) = \frac{x - 1}{x - 1} \) at \( x = 1 \)?


  It becomes undefined because both the numerator and denominator equal zero.
• What is the value of \( f(1) \)?


  Undefined
• How does the function \( f(x) \) behave for values of \( x \) other than 1?


  It equals 1 for all \( x \) except 1.
• What does the graph of \( f(x) \) look like around \( x = 1 \)?


  It is a horizontal line at \( y = 1 \) with a gap at \( x = 1 \).
• What does the limit as \( x \) approaches 1 of \( f(x) \) equal to?


  1
• What is the notation used to express the limit as \( x \) approaches 1 of \( f(x) \)?


  \( \lim_{x \to 1} f(x) = 1 \)
• What is the definition of the function \( g(x) \) as described in the video?


  \( g(x) = x^2 \) when \( x \neq 2 \) and \( g(2) = 1 \)
• How does the graph of \( g(x) \) behave at \( x = 2 \)?


  There is a gap at \( x = 2 \) because \( g(2) = 1 \).
• What is the value of \( g(2) \)?


  1
• What is the limit as \( x \) approaches 2 of \( g(x) \)?


  4
• How can you determine the limit of \( g(x) \) as \( x \) approaches 2 numerically?


  By evaluating \( g(x) \) at values close to 2, such as 1.9 and 2.1.
• What happens to \( g(x) \) as \( x \) approaches 2 from the left side?


  It approaches 4.
• What happens to \( g(x) \) as \( x \) approaches 2 from the right side?


  It also approaches 4.
• What are the key characteristics of limits in functions?


  • A limit describes the behavior of a function as it approaches a certain point.
  • Limits can exist even when the function is undefined at that point.
  • The limit can be evaluated from both the left and right sides.
• What are the differences between the functions \( f(x) \) and \( g(x) \)?


  • \( f(x) = \frac{x - 1}{x - 1} \) is undefined at \( x = 1 \) but equals 1 elsewhere.
  • \( g(x) = x^2 \) except at \( x = 2 \), where it equals 1.
  • \( f(x) \) has a gap at \( x = 1 \), while \( g(x) \) has a gap at \( x = 2 \).