## Finding Limits with Direct Substitution

Direct substitution is just what the name implies: you directly substitute a given value into a limit.

Probably the most intuitive way to find a limit is to look at a limit graphically (on a graphing calculator) or numerically (through a table). However, in some cases it’s actually easier—and faster—to find a limit with direct substitution. If you’ve ever put a specific value into an equation in algebra (like putting x = 2 into the function y = x + 10), then you have already performed direct substitution. It’s really that easy!

## When to Use Direct Substitution

Direct substitution works well when you have a simple functions with additions, subtractions, divisions, multiplications, powers and roots.

For example, this technique can work with constant functions, like:

- f(x) = 10,
- f(x) = 6,
- f(x) = 99.99.

You can also use it to find limits for linear functions, which produce straight lines in a graph; they have “x” as the input variable and the “x” has an exponent of one. Examples of linear functions are:

- f(x) = x,
- f(x) = 2x – 2,
- f(x) = x + 1.

**When NOT to use direct substitution**: If you plug in x-values (using the steps below) and get an indeterminate limit (either 0/0 or ∞/∞), you can’t use this technique. Use the dividing out technique for limits instead.

## Find Limits with Direct Substitution: Steps

## Example problem 1:

**Find the limit of f(x) = 9x – 2 at x = 6.**

Step 1: **Make sure the function as a simple function** (one of the types shown in the above image). This particular function is made up of *sums*, so we can use the rule.

Step 2: **Substitute the “x” value (the point at which you want to find the limit) into the function. **In this example, you’re looking for the limit at x = 6, so:

f(x) = 9x – 2

f(x) = 9(6) – 2

f(x) = 54 – 2

f(x) = 52

The limit of f(x) = 9x – 2 as x approaches 6 is 52.

A quick look at a graph of f(x) = 9x – 2 shows this is the correct answer:

## Example 2

Find the limit of f(x) = √(x) at x = 4

Step 1: **Make sure the function as a simple function**. This is a radical function, so we can use the rule.

Step 2: **Substitute the “x” value into the function. **In this example, you’re looking for the limit at x = 4, so:

√(4) = 2

The limit of f(x) = √(x) as x approaches 4 is 2.

## Example 3

This next example may look a little more complicated, but it isn’t: you’re still just “filling in the blanks.”

**Find the limit:**

Step 1: **Make sure the function as a simple function**. This is a radical function, so we can use the rule.

Step 2: **Substitute the “x” value into the function. **In this example, you’re looking for the limit at x = 6, so:

The limit as x approaches 6 is 4.

## Polynomials and Radicals

Direct substitution can also work for polynomial functions and radical functions, as long as you are sure the function is defined at the x-value you want to find the limit at. For example, you can use direct substitution for all values of f(x) = 1/x, except at 0 (because division by zero is undefined). If your algebra isn’t strong, a good rule of thumb is that you shouldn’t use direct substitution to solve for the limit unless you are **sure** that the function is defined for all real numbers.

## References

Berresford, G. & Rockett, A. (2015). Applied Calculus. Cengage Learning.