Limits Theorems Part 1

Limit Properties

• The limit of a constant is the constant itself: lim 4 = 4 regardless of the value of x.
• The limit of x as x approaches a value is the value itself: lim x = 2 as x → 2.

Limit Theorems

• The limit of a sum is the sum of the limits: lim (x + 7) = lim x + lim 7 = 4 + 7 = 11 as x → 4.
• The limit of a difference is the difference of the limits: lim (2x - 3 + 3x - 5) = lim 2x - 3 + lim 3x - 5 as x → 4.
• The limit of a product is the product of the limits: lim 2x = lim 2 lim x = 2(3) = 6 as x → 3.
• The limit of a composite function is the limit of the outer function times the limit of the inner function: lim 2x(3x - 1) = lim 2x lim (3x - 1) as x → 4.

Polynomial Function

• A polynomial function is defined as: f(x) = c0 + c1x + c2x^2 + … + cnx^n.
• The limit of a polynomial function as x approaches a is: lim f(x) = lim (c0 + c1x + c2x^2 + … + cnx^n) = f(a) as x → a.
• To evaluate the limit of a polynomial function, substitute a in f(x).

Evaluating Limits

• To evaluate a limit, substitute the value of x into the function: lim 2x^3 + 3x^2 - 1 = 2(2)^3 + 3(2)^2 - 1 = 27 as x → 2.
• To evaluate a limit, substitute the value of x into the function: lim 3x^7 - 5x^4 + 2x^3 + 9 = -1 as x → -1.