Limits · Grades 12–12
Limits Worksheets
Introductory limit problems solved by direct substitution, the first step into calculus, evaluating what a polynomial approaches as x nears a given value.
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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
Every derivative and integral you'll meet later is quietly built out of a limit, so mastering direct substitution now is a rehearsal for the idea calculus runs on before the notation gets harder.
How to do it
- Identify the value a that x is approaching.
- Substitute a for every x in the function.
- Evaluate the resulting expression, that number is the limit.
Examples by level
- Beginnerlim (x → −2) [3x² + 5] =Answer17
- Intermediatelim (x → 5) [2x² − 3] =Answer47
- Advancedlim (x → 3) [−9x³ + 6x² − 2] =Answer-191
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.
For parents
Frame it simply: 'what number does this expression get closer and closer to as x gets closer to this value?', for a polynomial, that's just plugging the number in.
For teachers
Keep this strictly to polynomials with no indeterminate forms (0/0) at this stage, that's a deliberately separate, harder skill for later.
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.
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