Let's take another look at some of the functions we have been exploring,
but using a table of values in addition to the graph of the function. Each
example includes a table of values of the function which approach *c* from the left and right.

See About the calculus applets for operating instructions. |

The first graph show the line used in a previous example. Is the limit *L* = 0.5 when *c* = 1 ? In other words, does *f *(*x*) approach 0.5 as *x* approaches 1? Looking at the
table of values, as *x* approaches 1 from either direction, the
output value of the function approaches 0.5. Note that the table does not
present a value for *f *(*c*) as this is not needed to find the
limit.

Select the second example. This is just like the first case, except
that one point has moved. Is the limit still *L* = 0.5 when *c* = 1 ? In other words, does *f *(*x*) approach 0.5 as *x* approaches 1? Looking at the table of values, as *x* approaches
1 from either direction, the output value of the function approaches 0.5.
The fact that *f *(*c*) does not equal 0.5 (it equals 1) has no
effect on the limit.

Select the third example. This is like the previous two cases, but
there is now a point missing. Is the limit still *L* = 0.5 when *c* = 1 ? In other words, does *f *(*x*) approach 0.5 as *x* approaches 1? Looking at the table of values, as *x* approaches 1 from either direction, the output value of the function
approaches 0.5. The fact that *f *(*c*) is undefined has no
effect on the limit.

Select the fourth example. This is a more complex function, but this
example is similar to the previous one with a missing point. What is the
limit when *c* = 0 ? In other words, what value does *f *(*x*) approach as *x* approaches 0? As you can see from the
table of values, the output value approaches 1 from both directions.

Select the fifth example, a jump discontinuity. What is the limit when *c* = 1? In other words, what value does *f *(*x*)
approach as *x* approaches 1? Note that in this case, the table
shows different values for the left-hand and right-hand limits. Hence
there is no general limit at *c* = 1.

Select the sixth example, a function with a vertical asymptote. What is
the limit when *c* = 1? In other words, what value does *f *(*x*) approach as *x* approaches 1? The table shows that
the output value gets bigger and bigger as you approach 1 from either
direction, hence there is no limit.

Select the seventh example, the wiggly *sin*(1/*x*). The
table shows that the value jumps around as you approach 0, hence the
limit does not exist there.

- Constant, Line, and Power Functions
- Exponential Functions
- Trigonometric Functions
- Constant Multiple
- Combinations: Sum and Difference
- Combinations: Product and Quotient
- Composition of Functions (the Chain Rule)
- Transformations of Functions
- Inverses of Functions
- Hyperbolic Functions
- Linear Approximation
- Mean Value Theorem

(C) 2011 Copyright Math Open Reference.

All rights reserved

All rights reserved