- Topics At a Glance
- Limits
- Functions Are Your Friend
- Graphing and Visualizing Limits
- Piecewise Functions and Limits
- One-Sided Limits
- Limits via Tables
- Limits via Algebra
- All About Asymptotes
- Vertical Asymptotes
- Finding Vertical Asymptotes
- Vertical Asymptotes vs. Holes
- Limits at Infinity
- Natural Numbers
- Limits Approaching Zero
- Estimating a Circle
- The Cantor Set and Fractals
- Limits of Functions at Infinity
- Horizontal, Slant, and Curvilinear Asymptotes
- Horizontal Asymptoes
- Slant Asymptotes
- Curvilinear Asymptotes
- Finding Horizontal/ Slant/ Curvilinear Asymptotes
- How to Draw Rational Functions from Scratch
- Comparing Functions
- Power Functions vs. Polynomials
- Polynomials vs. Logarithmic Functions
**Manipulating Limits**- The Basic Properties
**Multiplication by a Constant**- Adding and Subtracting Limits
- Multiplying and Dividing Limits
- Powers and Roots of Limits
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

If

,

then

.

This rule is often written

.

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

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.

Example 1

If \lim_x\to1 find \lim_x\to12 |

Example 2

If \lim_x\to24 find \lim_x\to2 |