Limits  Part I - Review from MAT265 

Limit Rules 

  1) limit(f(x)+g(x), x = a) = (limit(f(x), x = a))+(limit(g(x), x = a)) 

  2) limit(f(x)-g(x), x = a) = (limit(f(x), x = a))-(limit(g(x), x = a)) 

  3) limit(f(x)*g(x), x = a) = (limit(f(x), x = a))*(limit(g(x), x = a)) 

  4)  

  5) limit(c*f(x), x = a) = c*(limit(f(x), x = a)) 

  6)  

  7)f(x)^(1/n)  

  8)

 

  9)limit(c, x = a) = c 

Polynomials, P(x) 

  1) limit(P(x), x = a) = P(a) 

  2)  

Theorem (**) 


    Similarly,

L'Hopital's Rule 

          If

 

          then  

Squeeze Theorem 

      If   g(x) <= f(x) and f(x) <= h(x)    and limit(g(x), x = a) = Land 

      then limit(f(x), x = a) = L 

Rational Functions 

      If  

     then divide f(x)and g(x)by the highest power of x in the numerators, simplify, and use Theorem (**). 

    (Or you can use L'Hopital's Rule) 

Exponentials and Logarithms 

     limit(e^x, x = infinity) = infinity limit(ln(x), x = infinity) = infinity

Part II - New from MAT266 

limit(r^n, n = infinity) = piecewise(-1 < r and r < 1, 0, r = 1, 1) 

                          Diverges    otherwise