Can you draw an example of a function that:

1) Is defined everywhere, but does not have a limit at

2) Is not defined at , but does have a limit there

3) Has a limit at x=c that is not the same value as

1.3: Evaluating Limits Analytically

Limitations live only in our minds. But if we use our imaginations, our possibilities become limitless.

Properties of Limits

See Theorem 1.1 and 1.2

1. Limits of Polynomials lim ( ) ( )x c

p x p c

2

3Ex: lim 2 5

xx x

1) Try direct substitution first, if real number then done!

1. Limits of Rational Functions

2 2 8( )

2

x xf x

x

2) If direct substitution fails:

Simplify to find another function that agrees for all x, except x = c

2 2

1

2 8 (1) 2(1) 8 5lim 5

2 1 2 1x

x x

x

lim𝑥→1

𝑓 (𝑥)Find:

Evaluate the following limits.2

24

5 41) lim

2 8x

x x

x x

3

1 22) lim

3x

x

x

0

1 14 43) lim

x

xx

Summary – to find limits analytically

1) Try direct substitution

2) Simplify to a single fraction

3) Try factoring/cancel terms

4) Multiply by conjugate

Note: If none of these work, a graph or numerical investigation may give you insight into whether the limit exists at all (recall the 3 cases!)