Limits by direct substitution worksheet for Precalculus

Precalculus introduces limits near the end of the course, and direct substitution is the first tool. You use it on polynomials, on rational functions whose bottom is not zero, and on functions with trig, e and ln, and you combine limits given as numbers with the limit laws.

Also taught in Calculus, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Limits by Direct Evaluation Date Period Evaluate each limit, or answer as asked. Evaluate limits by direct substitution $\lim_{x \to 3} (8 x^{2} - 4)$ $\lim_{x \to \frac{π}{4}} (\sec^{2} x - \tan x)$ $\lim_{x \to 0} \sqrt{x + 2}$ $\lim_{x \to 5} \frac{2 x^{2} - x - 1}{x^{2} + 5 x}$ $\lim_{x \to π} (\frac{\cos x}{1 + x})$ $\lim_{x \to \frac{5 π}{6}} \sec x$ $\lim_{x \to 5} \frac{- 8 x - 7}{4 x^{2} + 5}$ $\lim_{x \to \frac{π}{4}} (\sin x \cos x)$ $\lim_{x \to \frac{3 π}{2}} \cos \frac{x}{2}$ $\lim_{x \to - 3} 2 \sqrt{x + 7}$ $\lim_{x \to \frac{π}{2}} (x \sin x)$ $\lim_{x \to 5} \sqrt{3 x + 7}$ $\lim_{x \to 5} (5 x^{3} + 6 x^{2} - 5)$ $\lim_{x \to 1} (x^{2} \ln x + 3)$ $\lim_{x \to - 3} \frac{x^{2} + 3}{x - 4}$ Use the limit laws If $\lim_{x \to 4} f (x) = - 8$ and $\lim_{x \to 4} g (x) = 12$ , find $\lim_{x \to 4} \sqrt{f (x) + g (x)}$ If $\lim_{x \to 5} f (x) = - 7$ and $\lim_{x \to 5} g (x) = - 8$ , find $\lim_{x \to 5} [x^{2} f (x) - g (x)]$ If $\lim_{x \to 1} f (x) = - 8$ and $\lim_{x \to 1} g (x) = - 3$ , find $\lim_{x \to 1} \frac{2 f (x)}{g (x) - 3}$ If $\lim_{x \to - 2} f (x) = 8$ and $\lim_{x \to - 2} g (x) = - 4$ , find $\lim_{x \to - 2} [(f (x))^{2} - 2 g (x)]$ If $\lim_{x \to - 5} f (x) = - 5$ and $\lim_{x \to - 5} g (x) = 4$ , find $\lim_{x \to - 5} [x^{2} f (x) - g (x)]$ 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, or answer as asked.

Evaluate limits by direct substitution
$\lim_{x \to 3} (8 x^{2} - 4)$
$\lim_{x \to \frac{π}{4}} (\sec^{2} x - \tan x)$
$\lim_{x \to 0} \sqrt{x + 2}$
$\lim_{x \to 5} \frac{2 x^{2} - x - 1}{x^{2} + 5 x}$
$\lim_{x \to π} (\frac{\cos x}{1 + x})$
$\lim_{x \to \frac{5 π}{6}} \sec x$
$\lim_{x \to 5} \frac{- 8 x - 7}{4 x^{2} + 5}$
$\lim_{x \to \frac{π}{4}} (\sin x \cos x)$
$\lim_{x \to \frac{3 π}{2}} \cos \frac{x}{2}$
$\lim_{x \to - 3} 2 \sqrt{x + 7}$
$\lim_{x \to \frac{π}{2}} (x \sin x)$
$\lim_{x \to 5} \sqrt{3 x + 7}$
$\lim_{x \to 5} (5 x^{3} + 6 x^{2} - 5)$
$\lim_{x \to 1} (x^{2} \ln x + 3)$
$\lim_{x \to - 3} \frac{x^{2} + 3}{x - 4}$

Use the limit laws
If $\lim_{x \to 4} f (x) = - 8$ and $\lim_{x \to 4} g (x) = 12$ , find $\lim_{x \to 4} \sqrt{f (x) + g (x)}$
If $\lim_{x \to 5} f (x) = - 7$ and $\lim_{x \to 5} g (x) = - 8$ , find $\lim_{x \to 5} [x^{2} f (x) - g (x)]$
If $\lim_{x \to 1} f (x) = - 8$ and $\lim_{x \to 1} g (x) = - 3$ , find $\lim_{x \to 1} \frac{2 f (x)}{g (x) - 3}$
If $\lim_{x \to - 2} f (x) = 8$ and $\lim_{x \to - 2} g (x) = - 4$ , find $\lim_{x \to - 2} [(f (x))^{2} - 2 g (x)]$
If $\lim_{x \to - 5} f (x) = - 5$ and $\lim_{x \to - 5} g (x) = 4$ , find $\lim_{x \to - 5} [x^{2} f (x) - g (x)]$



Sigma Prep · sigmaprep.io/worksheets Name Limits by Direct EvaluationAnswer keyVersion 2 Date Period Evaluate each limit, or answer as asked. Evaluate limits by direct substitution $\lim_{x \to 3} (8 x^{2} - 4)$ $68$ $\lim_{x \to \frac{π}{4}} (\sec^{2} x - \tan x)$ $1$ $\lim_{x \to 0} \sqrt{x + 2}$ $\sqrt{2}$ $\lim_{x \to 5} \frac{2 x^{2} - x - 1}{x^{2} + 5 x}$ $\frac{22}{25}$ $\lim_{x \to π} (\frac{\cos x}{1 + x})$ $- \frac{1}{1 + π}$ $\lim_{x \to \frac{5 π}{6}} \sec x$ $- \frac{2 \sqrt{3}}{3}$ $\lim_{x \to 5} \frac{- 8 x - 7}{4 x^{2} + 5}$ $- \frac{47}{105}$ $\lim_{x \to \frac{π}{4}} (\sin x \cos x)$ $\frac{1}{2}$ $\lim_{x \to \frac{3 π}{2}} \cos \frac{x}{2}$ $- \frac{\sqrt{2}}{2}$ $\lim_{x \to - 3} 2 \sqrt{x + 7}$ $4$ $\lim_{x \to \frac{π}{2}} (x \sin x)$ $\frac{π}{2}$ $\lim_{x \to 5} \sqrt{3 x + 7}$ $\sqrt{22}$ $\lim_{x \to 5} (5 x^{3} + 6 x^{2} - 5)$ $770$ $\lim_{x \to 1} (x^{2} \ln x + 3)$ $3$ $\lim_{x \to - 3} \frac{x^{2} + 3}{x - 4}$ $- \frac{12}{7}$ Use the limit laws If $\lim_{x \to 4} f (x) = - 8$ and $\lim_{x \to 4} g (x) = 12$ , find $\lim_{x \to 4} \sqrt{f (x) + g (x)}$ $2$ If $\lim_{x \to 5} f (x) = - 7$ and $\lim_{x \to 5} g (x) = - 8$ , find $\lim_{x \to 5} [x^{2} f (x) - g (x)]$ $- 167$ If $\lim_{x \to 1} f (x) = - 8$ and $\lim_{x \to 1} g (x) = - 3$ , find $\lim_{x \to 1} \frac{2 f (x)}{g (x) - 3}$ $\frac{8}{3}$ If $\lim_{x \to - 2} f (x) = 8$ and $\lim_{x \to - 2} g (x) = - 4$ , find $\lim_{x \to - 2} [(f (x))^{2} - 2 g (x)]$ $72$ If $\lim_{x \to - 5} f (x) = - 5$ and $\lim_{x \to - 5} g (x) = 4$ , find $\lim_{x \to - 5} [x^{2} f (x) - g (x)]$ $- 129$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits by Direct EvaluationAnswer keyVersion 2

Date Period

Evaluate each limit, or answer as asked.

Evaluate limits by direct substitution
$\lim_{x \to 3} (8 x^{2} - 4)$ $68$
$\lim_{x \to \frac{π}{4}} (\sec^{2} x - \tan x)$ $1$
$\lim_{x \to 0} \sqrt{x + 2}$ $\sqrt{2}$
$\lim_{x \to 5} \frac{2 x^{2} - x - 1}{x^{2} + 5 x}$ $\frac{22}{25}$
$\lim_{x \to π} (\frac{\cos x}{1 + x})$ $- \frac{1}{1 + π}$
$\lim_{x \to \frac{5 π}{6}} \sec x$ $- \frac{2 \sqrt{3}}{3}$
$\lim_{x \to 5} \frac{- 8 x - 7}{4 x^{2} + 5}$ $- \frac{47}{105}$
$\lim_{x \to \frac{π}{4}} (\sin x \cos x)$ $\frac{1}{2}$
$\lim_{x \to \frac{3 π}{2}} \cos \frac{x}{2}$ $- \frac{\sqrt{2}}{2}$
$\lim_{x \to - 3} 2 \sqrt{x + 7}$ $4$
$\lim_{x \to \frac{π}{2}} (x \sin x)$ $\frac{π}{2}$
$\lim_{x \to 5} \sqrt{3 x + 7}$ $\sqrt{22}$
$\lim_{x \to 5} (5 x^{3} + 6 x^{2} - 5)$ $770$
$\lim_{x \to 1} (x^{2} \ln x + 3)$ $3$
$\lim_{x \to - 3} \frac{x^{2} + 3}{x - 4}$ $- \frac{12}{7}$

Use the limit laws
If $\lim_{x \to 4} f (x) = - 8$ and $\lim_{x \to 4} g (x) = 12$ , find $\lim_{x \to 4} \sqrt{f (x) + g (x)}$ $2$
If $\lim_{x \to 5} f (x) = - 7$ and $\lim_{x \to 5} g (x) = - 8$ , find $\lim_{x \to 5} [x^{2} f (x) - g (x)]$ $- 167$
If $\lim_{x \to 1} f (x) = - 8$ and $\lim_{x \to 1} g (x) = - 3$ , find $\lim_{x \to 1} \frac{2 f (x)}{g (x) - 3}$ $\frac{8}{3}$
If $\lim_{x \to - 2} f (x) = 8$ and $\lim_{x \to - 2} g (x) = - 4$ , find $\lim_{x \to - 2} [(f (x))^{2} - 2 g (x)]$ $72$
If $\lim_{x \to - 5} f (x) = - 5$ and $\lim_{x \to - 5} g (x) = 4$ , find $\lim_{x \to - 5} [x^{2} f (x) - g (x)]$ $- 129$

How to do these

If $\lim_{x \to 1} f (x) = 4$ and $\lim_{x \to 1} g (x) = 2$ , find $\lim_{x \to 1} [2 f (x) - g {(x)}^{2}]$

The limit of a difference is the difference of the limits.
2 times 4 is 8, and 2 squared is 4.
8 - 4.

The answer is $4$ .

Where students go wrong

Substituting where the function is not defined, such as ln x at x = 0 or tan x at π/2. Direct substitution only works where the function is continuous, so check the domain before trusting the number.

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

What you can put on this worksheet

Substituting straight in: $\lim_{x \to 0} (- 3 x^{2} + 3 x - 5)$ → $- 5$
Limit laws from given limits: If $\lim_{x \to - 1} f (x) = 9$ and $\lim_{x \to - 1} g (x) = - 6$ , find $\lim_{x \to - 1} [f (x) g (x)]$ → $- 54$
Trig, e and ln: $\lim_{x \to π} (\frac{\cos x}{1 + x})$ → $- \frac{1}{1 + π}$
Through a cube root: $\lim_{x \to - 5} (- 2 \sqrt[3]{3 x + 23} - 9)$ → $- 13$
Constants, fractions, roots and angles past the first quadrant: $\lim_{x \to 0} \frac{x^{2} + 1}{x + 1}$ → $1$

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.

Precalculus is usually taken around 11th or 12th grade, 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.

You pick from 5: substituting straight in, limit laws from given limits, trig, e and ln, through a cube root and constants, fractions, roots and angles past the first quadrant. Tick as many as you want and set how many of each, or let it spread them evenly.

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 for Precalculus

Select your difficulty

Evaluate limits by direct substitution

Use the limit laws

Evaluate limits by direct substitution

Use the limit laws

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these limits by direct substitution worksheets free?+

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Are these limits by direct substitution worksheets printable?+

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Is this a free alternative to Kuta Software for limits by direct substitution?+

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What do students get wrong with limits by direct substitution?+