Limits

Limits

1. Properties of Limits

   • Use: Lim f(x)= L lim g(x)=k x → c x → c
     • 1) Scalar Multiple: lim (bf(x))= bL x→ c
       • 2) Sum/difference: lim [f(x) _+ g(x)] = L _+ k x→ c
         • 3) Product: lim [f(x) g(x)] = Lk x→ c
           • 4) Quotient: lim f(x)/g(x) = L/k, k ≠ 0 x→ c
             • 5) lim [f(x)]^n = L^n x→c

2. Limits at Infinite

   • Use: lim f(x) = ∞ lim g(x)= L x→ c x→ c
     • 1) lim [f(x) + g(x)] = ∞ x→c
       • 2) lim [b f(x)] ∞, b>0 or ∞, b<0 x→c
         • 3) lim [g(x)/f(x)] = 0 x→c

3. Continuity

   • Use: A function is continuos at point x=c, if:
     • 1) f(c) is defined
       • 2) lim f(x) exists x→c
         • 3) f(c)= lim f(x) x→c
           • A hole is considered a removable discontinuity
             • An asymptote or jump is a non-removable discontinuity

4. Limits

   • A limit is an approximation of f(x) when the x approaches to a number, always the value of y.
     • x-1/x-1 → substitute If f(1) is undefined, then use limits.
       • If we approach to the same value of the function, we have a limit in that value.
         • But, if the value of f(x) is DIFFERENT, the limit DOES NOT EXIST

5. Limits Numerically

   • Ex: lim x^3-1/x-1 → x=1 =0/0 = undefined
     • Do a table of values putting the numbers that are closest to x=n from the left and from the right, and with those numbers substitute in the equation and plug in the table of values.

6. Limits Algebraically

   • First plug the value in the equation, if you get another value, that is the answer. Although, if you get undefined, then you FACTOR.
     • a^2+b^2= (a+b) (a+b)
       • a^3 + b^3= (a^2 -ab +b^2)
         • a^3 - b^3= a^2 + ab + b^2

7. Using Conjugate

   • When the equation cannot be factored, then you have to multiply times the conjugate of that equation.
     • Conjugate: same equation, but the signs in the middle become the opposite of the original sign.
       • Example: √x -3/ x-9. Conjugate: √x + 3/√x + 3

8. Derivatives

   • A derivative is the SLOPE of a TANGENT line
     • Formula: f (x + h) -f (x)/ x + h
       • Step 1: Copy the formula (below)
         • Step 2: Evaluate, substitute all the x's of the original equation by (x + h)
           • Copy the second part of the equation with the plugged values
             • Step 3: substitute h by 0 since h → 0
               • Finish evaluating
   • Can be: f'(x)
     • y'
       • dy/dx

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