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Functions, Graphs, and Limits

Functions, Graphs, and Limits

At a Glance - Multiplication by a Constant

If

 ,

then

 .

This rule is often written

 .

We say that we're "pulling out'' the constant c from the limit.

Sample Problem

For example,

and

 .

Be Careful: This rule is only valid if

is actually defined and equals L for some real number L.

We wouldn't say

because what does it mean to multiply 3 by infinity? That's like saying 3 × undefined, which doesn't make sense.

If   ,

is undefined (including if it equals ∞ or -∞), then the limit

is also undefined.

In pictures, if we multiply a function by a constant it means we're stretching or shrinking the function vertically. We can also stretch or shrink the limit.

For example, take the line f(x) = x and see what happens if we multiply it by 3:

As the function gets stretched, so does the limit. If we originally had

then as we stretch the function by a factor of 3, the limit will also be stretched by a factor of 3:

If we shrink the function by , the limit will shrink by the same factor:

The limit will go from

to

Sometimes we may be asked to find a limit given partial information about a function.

For example if we're given that limxc f(x) = 4, then no matter what function f is,

limxcf(x) = 3·limxc f(x) = 3 · 4 = 12.

Example 1

If limx → 1 f(x) = 4, what is limx → 1f(x)?


Example 2

If limx → 2f(x) = 12,

what is limx → 2 f(x)?


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