Limits Definition and Properties

Limits

Definition of limit

The definition of a limit is the value a function f(x) approaches as x gets sufficiently close to a specified value from both directions, from the left and right.

Form this graph, we can see that as receives close to a, from both directions, the given function approaches b. In limit notation, this can write as   = b  --> indicate (lim_(x->a) f(x) = b).


and read as , the limit of f(x) is b as x approaches a.

Properties

Let a,b and r be real numbers, let lim_(x->a) f(x) = A and lim_(x->a) g(x) = B


 (1) lim_(x->a) bf(x) = b*lim_(x->a) f(x)  --> Scalar Multiple

 (2) lim_(x->a) [f(x) ± g(x)] = lim_(x->a) f(x) ± lim_(x->a) g(x) = A ± B  -->  Sum and Difference

 (3) lim_(x->a) f(x)g(x) = lim_(x->a) f(x) * lim_(x->a) g(x) = AB --> Product

 (4) lim_(x->a) f(x)/g(x) = lim_(x->a) f(x)/lim_(x->a) g(x) = A/B (g(x) ≠ 0, B ≠ 0) --> Quotient

 (5) lim_(x->a) [f(x)]^r = A^r --> Exponent
      (^ means exponent  )