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# Functions, Graphs, and Limits

# 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 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.