Limits Worksheets

What it is

A limit describes what value a function approaches as x gets close to some number a. For a polynomial, which has no breaks or holes, the limit as x approaches a is simply the function's value AT a, found by direct substitution.

Why it matters

Limits are the foundational idea calculus is built on, derivatives and integrals are both defined in terms of limits. Mastering direct substitution on polynomials (the simplest case) builds the intuition needed before tackling harder limit forms.

How to do it

1. Identify the value a that x is approaching.
2. Substitute a for every x in the function.
3. Evaluate the resulting expression, that number is the limit.

Examples by level

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

• Treating the limit as if it needs the function to be defined AT that exact point in some special way, rather than just substituting.
• Arithmetic errors when substituting negative values, forgetting to square or cube the sign correctly.
• Leaving the answer as an expression instead of a single evaluated number.

Tips

• For polynomials, direct substitution always works, there's no need to check for holes or asymptotes like with rational functions.
• The limit is a single NUMBER, not an expression, keep substituting and simplifying until you reach one.

Key vocabulary

limit

the value a function approaches as the input gets close to a given number

direct substitution

plugging the target value straight into the function to evaluate the limit

Frequently asked questions

What grade are limits taught in?

Limits are typically the opening topic of a Grade 12 calculus course, introduced before derivatives.

Is a limit the same as the function's value?

For a polynomial, yes, polynomials have no breaks, so the limit always equals direct substitution. This isn't true for all functions, but it's the right starting point.