Basic Calculus

• Steps to estimate the limit of a function as x approaches a

  1. Substitute the value of a into the function

  2. If the result is a real number, then the limit is that real number

  3. If the result is an indeterminate form (e.g. 0/0, ∞/∞), then additional techniques are needed to evaluate the limit
• Determining the limit of a function without graphing or using a table of values

  Apply the different limit laws (e.g. limit of a constant, limit of an identity function, constant multiple law, sum/difference law, product law, quotient law, power law, root law)
• The Basic Limit Laws

  • Limit of a constant function
  • Limit of the identity function
  • Constant multiple law
  • Sum or difference law
  • Product law
  • Quotient law
  • Power law
  • Root law
• Limit of a constant function

  The limit of a constant function is the constant itself. Let k be any constant, then:

  lim k=k

  x->c
• Limit of the identity function

  The limit of the identity function f(x) = x as x approaches a is equal to c

  lim x=c

  x->c
• Constant multiple law

  The limit of a constant k multiplied by a function f(x) is equal to k multiplied by the limit of the function.

  lim k ⋅ f(x) = k ⋅ lim f(x) = k ⋅ L

  x->c x->c
• Sum or difference law

  The limit of the sum or difference of two functions is equal to the sum or difference of the limits of the two functions

  lim [f(x) ± g(x)] = lim f(x) ± lim g (x) = L+M

  x->c x->c x->c
• Product law

  The limit of the product of two functions is equal to the product of the limits of the two functions

  lim[f(x)⋅g(x)] = lim f(x) ⋅ lim g(x) = L⋅M

  x->c x->c x->c
• Quotient law

  The limit of the quotient of two functions is equal to the quotient of the limits of the two functions, provided that the limit of the divisor is not equal to zero lim g(x) ≠ 0

  
  

  lim f(x)/g(x) = lim f(x)/lim g(x) = L/M

  x->c x->c x->c

  Provided that M≠0
• Power law

  The limit of the integral power of a function is equal to the integral power of the limit of the function, provided that the limit of the function is not equal to zero when the exponent is negative

  
  

  lim[f(x)]^n = [lim f(x)]^n = L^n

  x->c x->c
• Root law

  The limit of the n^th root of a function is equal to the n^th root of the limit of the function, where n is a positive integer, and the limit of the function is positive when n is even

  
  

  lim ^2√f(x) = ^2√lim f(x) = ^n√L

  x->c x->c

  and lim f(x) = L > 0 when n is even

  x->c
• Direct substitution may not be applicable to some limits that will be discussed in the next lesson
• Using the limit laws, we can prove that lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x) given that the limits lim(x→a) f(x) and lim(x→a) g(x) both exist