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Unit 7: Limits

7.1 Limits and Function Values Notes

If the limit of a function f as x approaches c exists, this limit may not be equal to f(c). In fact, f(c)

may not even be defined.
Non-existence of Limits

The limit of a function f as x approaches c may fail to exist if:
f(x) becomes infinitely large or infinitely small as x approaches c from either side.

f(x) approaches L as x approaches c from the right and f(x) approaches M, M L, as x
approaches c from the left.

f(x) oscillates infinitely many times between two numbers as x approaches c from either side.
Limit of a Constant

If d is a constant, then = d.

Limit of the Identity Function
For every real number c, = c

7.1.1 Properties of Limits

If f and g are functions and c, L, and M are numbers such that and then
=

= L + M
=

= L – M
=

= L ∙ M
=

= L/M, M 0
= f(x) 0 for all x near c.

Limits of Polynomial Functions

If f(x)is a polynomial function and c is any real number, then f(x) = f(c). In other words, the

limit is the value of the polynomial function f at x = c.

Limits of Rational Functions

Let f(x) be a rational function and let c be a real number such that f(c) is defined. Then f(x)
= f(c).

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