Limits by direct substitution worksheet

Where a function is continuous, the limit is simply its value there, so you put the number in and read off the answer. Knowing when that is allowed is the point: it fails exactly where the function is not defined.

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Sigma Prep · sigmaprep.io/worksheets Name Limits by Direct Evaluation Date Period Evaluate each limit. $\lim_{x \to 1} 7 x^{3} - 3 x^{2} + 4 x + 6$ $\lim_{x \to - 3} - 3 x^{2} - x$ $\lim_{x \to - 1} \frac{- 2 x^{2} + 3 x + 5}{2 x^{2} - 3 x}$ $\lim_{x \to - 3} x^{3} + 6 x^{2} - 5$ $\lim_{x \to - 1} 7 x^{3} - 7 x^{2} + 6$ $\lim_{x \to - 2} \frac{5 x^{2} + x + 6}{- 3 x + 4}$ $\lim_{x \to 0} 2 x^{3} - x^{2} - 5 x - 7$ $\lim_{x \to - 4} 5 x^{2} + 2 x + 4$ $\lim_{x \to 5} \frac{- 2 x^{2} + 2}{- 2 x^{2}}$ $\lim_{x \to 3} - 3 x^{2} + 7 x - 7$ $\lim_{x \to 1} \frac{- 2 x - 6}{- 2 x^{2} - x - 5}$ $\lim_{x \to 4} 3 x^{3} - 5 x^{2} + 6 x - 2$ $\lim_{x \to 2} - 7 x^{2} - 7 x - 3$ $\lim_{x \to 0} \frac{- x^{2} + 2 x}{- 4 x^{2} - 2 x + 1}$ $\lim_{x \to 2} - 6 x^{3} + x^{2} - 7$ $\lim_{x \to - 1} \frac{- 5 x^{2} + x - 6}{2 x + 4}$ $\lim_{x \to - 5} - 5 x^{2} + x + 5$ $\lim_{x \to 4} \frac{4 x - 1}{3 x + 4}$ $\lim_{x \to 2} \frac{x + 2}{x^{2} + 1}$ $\lim_{x \to 5} 5 x^{3} - 5 x^{2} - 7 x + 4$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits by Direct Evaluation

Date Period

Evaluate each limit.

$\lim_{x \to 1} 7 x^{3} - 3 x^{2} + 4 x + 6$
$\lim_{x \to - 3} - 3 x^{2} - x$
$\lim_{x \to - 1} \frac{- 2 x^{2} + 3 x + 5}{2 x^{2} - 3 x}$
$\lim_{x \to - 3} x^{3} + 6 x^{2} - 5$
$\lim_{x \to - 1} 7 x^{3} - 7 x^{2} + 6$
$\lim_{x \to - 2} \frac{5 x^{2} + x + 6}{- 3 x + 4}$
$\lim_{x \to 0} 2 x^{3} - x^{2} - 5 x - 7$
$\lim_{x \to - 4} 5 x^{2} + 2 x + 4$
$\lim_{x \to 5} \frac{- 2 x^{2} + 2}{- 2 x^{2}}$
$\lim_{x \to 3} - 3 x^{2} + 7 x - 7$
$\lim_{x \to 1} \frac{- 2 x - 6}{- 2 x^{2} - x - 5}$
$\lim_{x \to 4} 3 x^{3} - 5 x^{2} + 6 x - 2$
$\lim_{x \to 2} - 7 x^{2} - 7 x - 3$
$\lim_{x \to 0} \frac{- x^{2} + 2 x}{- 4 x^{2} - 2 x + 1}$
$\lim_{x \to 2} - 6 x^{3} + x^{2} - 7$
$\lim_{x \to - 1} \frac{- 5 x^{2} + x - 6}{2 x + 4}$
$\lim_{x \to - 5} - 5 x^{2} + x + 5$
$\lim_{x \to 4} \frac{4 x - 1}{3 x + 4}$
$\lim_{x \to 2} \frac{x + 2}{x^{2} + 1}$
$\lim_{x \to 5} 5 x^{3} - 5 x^{2} - 7 x + 4$



Sigma Prep · sigmaprep.io/worksheets Name Limits by Direct EvaluationAnswer keyVersion 1 Date Period Evaluate each limit. $\lim_{x \to 1} 7 x^{3} - 3 x^{2} + 4 x + 6$ $14$ $\lim_{x \to - 3} - 3 x^{2} - x$ $- 24$ $\lim_{x \to - 1} \frac{- 2 x^{2} + 3 x + 5}{2 x^{2} - 3 x}$ $0$ $\lim_{x \to - 3} x^{3} + 6 x^{2} - 5$ $22$ $\lim_{x \to - 1} 7 x^{3} - 7 x^{2} + 6$ $- 8$ $\lim_{x \to - 2} \frac{5 x^{2} + x + 6}{- 3 x + 4}$ $\frac{12}{5}$ $\lim_{x \to 0} 2 x^{3} - x^{2} - 5 x - 7$ $- 7$ $\lim_{x \to - 4} 5 x^{2} + 2 x + 4$ $76$ $\lim_{x \to 5} \frac{- 2 x^{2} + 2}{- 2 x^{2}}$ $\frac{24}{25}$ $\lim_{x \to 3} - 3 x^{2} + 7 x - 7$ $- 13$ $\lim_{x \to 1} \frac{- 2 x - 6}{- 2 x^{2} - x - 5}$ $1$ $\lim_{x \to 4} 3 x^{3} - 5 x^{2} + 6 x - 2$ $134$ $\lim_{x \to 2} - 7 x^{2} - 7 x - 3$ $- 45$ $\lim_{x \to 0} \frac{- x^{2} + 2 x}{- 4 x^{2} - 2 x + 1}$ $0$ $\lim_{x \to 2} - 6 x^{3} + x^{2} - 7$ $- 51$ $\lim_{x \to - 1} \frac{- 5 x^{2} + x - 6}{2 x + 4}$ $- 6$ $\lim_{x \to - 5} - 5 x^{2} + x + 5$ $- 125$ $\lim_{x \to 4} \frac{4 x - 1}{3 x + 4}$ $\frac{15}{16}$ $\lim_{x \to 2} \frac{x + 2}{x^{2} + 1}$ $\frac{4}{5}$ $\lim_{x \to 5} 5 x^{3} - 5 x^{2} - 7 x + 4$ $469$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits by Direct EvaluationAnswer keyVersion 1

Date Period

Evaluate each limit.

$\lim_{x \to 1} 7 x^{3} - 3 x^{2} + 4 x + 6$ $14$
$\lim_{x \to - 3} - 3 x^{2} - x$ $- 24$
$\lim_{x \to - 1} \frac{- 2 x^{2} + 3 x + 5}{2 x^{2} - 3 x}$ $0$
$\lim_{x \to - 3} x^{3} + 6 x^{2} - 5$ $22$
$\lim_{x \to - 1} 7 x^{3} - 7 x^{2} + 6$ $- 8$
$\lim_{x \to - 2} \frac{5 x^{2} + x + 6}{- 3 x + 4}$ $\frac{12}{5}$
$\lim_{x \to 0} 2 x^{3} - x^{2} - 5 x - 7$ $- 7$
$\lim_{x \to - 4} 5 x^{2} + 2 x + 4$ $76$
$\lim_{x \to 5} \frac{- 2 x^{2} + 2}{- 2 x^{2}}$ $\frac{24}{25}$
$\lim_{x \to 3} - 3 x^{2} + 7 x - 7$ $- 13$
$\lim_{x \to 1} \frac{- 2 x - 6}{- 2 x^{2} - x - 5}$ $1$
$\lim_{x \to 4} 3 x^{3} - 5 x^{2} + 6 x - 2$ $134$
$\lim_{x \to 2} - 7 x^{2} - 7 x - 3$ $- 45$
$\lim_{x \to 0} \frac{- x^{2} + 2 x}{- 4 x^{2} - 2 x + 1}$ $0$
$\lim_{x \to 2} - 6 x^{3} + x^{2} - 7$ $- 51$
$\lim_{x \to - 1} \frac{- 5 x^{2} + x - 6}{2 x + 4}$ $- 6$
$\lim_{x \to - 5} - 5 x^{2} + x + 5$ $- 125$
$\lim_{x \to 4} \frac{4 x - 1}{3 x + 4}$ $\frac{15}{16}$
$\lim_{x \to 2} \frac{x + 2}{x^{2} + 1}$ $\frac{4}{5}$
$\lim_{x \to 5} 5 x^{3} - 5 x^{2} - 7 x + 4$ $469$

How to do these

$\lim_{x \to 2} (x^{2} - 3 x + 1)$

The function is a polynomial, so it is continuous everywhere.
Put 2 in: 4 minus 6 plus 1.
That is the limit.

The answer is $- 1$ .

Where students go wrong

Substituting into a fraction without checking the bottom. If the denominator comes out zero, substitution has told you nothing and there is real work to do.

Also called direct substitution, evaluating limits or limits of polynomials.

Questions about these worksheets

Yes. Every limits by direct substitution sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the limits by direct substitution questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Substituting into a fraction without checking the bottom. If the denominator comes out zero, substitution has told you nothing and there is real work to do.

Limits by direct substitution worksheet

Select your difficulty

Limits by Direct Evaluation

Limits by Direct EvaluationAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

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