Adaptive practice for this skillA separate, individually graded pool that adapts to your answers: 24 validated questions, plus generated practice on the same skill.

Picture it first

Interactive · drag the slider

Watch the limit happen

Drag x toward 3. The point climbs the line toward the open hole at (3, 6) — and never has to reach it.

x = 23 · the limit pointx = 4

2.500

f(x)

5.500

gap to 6

0.500

You are approaching from the left. Keep sliding toward 3 and watch f(x) close in on 6.

x → 3⁻	f(x)	x → 3⁺	f(x)
2.9	5.9	3.1	6.1
2.99	5.99	3.01	6.01
2.999	5.999	3.001	6.001
↓ both columns close in on 6 ↓

Grade 12 · Practice problems

Limits

30 written problems across three levels — 10 foundational · 12 intermediate · 8 advanced. Work through them on paper, then reveal every answer to check. These are not graded and are counted separately from adaptive practice.

Foundational

1. FoundationalEvaluate $\lim_{x\to-1}(2x^3-x+4)$ .

2. FoundationalEvaluate $\lim_{x\to4}(x^2-5x+2)$ .

3. FoundationalEvaluate $\lim_{x\to0}\dfrac{x^2-3x}{x}$ .

4. FoundationalEvaluate $\lim_{x\to3}\dfrac{x^2-9}{x-3}$ .

5. FoundationalEvaluate $\lim_{x\to5}\dfrac{x-5}{x^2-25}$ .

6. FoundationalState the three conditions for continuity at $x=a$ .

7. FoundationalIf the left-hand limit is $2$ and the right-hand limit is $5$ , determine the two-sided limit.

8. FoundationalClassify a discontinuity whose finite one-sided limits are unequal.

9. FoundationalFor $f(x)=x+1$ when $x<2$ , evaluate $\lim_{x\to2^-}f(x)$ .

10. FoundationalA graph has a hole at $(4,7)$ and a filled point at $(4,1)$ , approaching the hole from both sides. Find $\lim_{x\to4}f(x)$ and $f(4)$ .

Intermediate

11. IntermediateEvaluate $\lim_{x\to0}\dfrac{\sqrt{x+9}-3}{x}$ .

12. IntermediateEvaluate $\lim_{x\to2}\dfrac{x^2-4}{x^2-x-2}$ .

13. IntermediateEvaluate $\lim_{x\to1}\dfrac{x^3-1}{x-1}$ .

14. IntermediateEvaluate $\lim_{x\to-2}\dfrac{x^2-4}{x+2}$ .

15. IntermediateEvaluate $\lim_{x\to0}\dfrac{(x+3)^2-9}{x}$ .

16. IntermediateEvaluate $\lim_{x\to4}\dfrac{\sqrt{x}-2}{x-4}$ .

17. IntermediateEvaluate $\lim_{x\to0}\dfrac{1}{x}\left(\dfrac{1}{x+2}-\dfrac{1}{2}\right)$ .

18. IntermediateEvaluate $\lim_{x\to2}\dfrac{x-2}{|x-2|}$ , or state that it does not exist.

19. IntermediateEvaluate $\lim_{x\to0}x^2\cos(3/x)$ using the squeeze theorem.

20. IntermediateWhat value of $f(2)$ makes $f(x)=\dfrac{x^2-4}{x-2}$ (for $x\ne2$ ) continuous at $x=2$ ?

21. IntermediateEvaluate $\lim_{x\to3^+}\dfrac{1}{x-3}$ and classify the discontinuity at $x=3$ .

22. IntermediateFor $f(x)=x+2$ when $x<1$ and $f(x)=5-x$ when $x\ge1$ , evaluate $\lim_{x\to1}f(x)$ .

Advanced

23. AdvancedEvaluate $\lim_{x\to0}\dfrac{\sqrt{1+x}-\sqrt{1-x}}{x}$ .

24. AdvancedFind $a$ so that $f(x)=x^2+1$ for $x<2$ and $f(x)=ax-3$ for $x\ge2$ is continuous at $x=2$ .

25. AdvancedUsing one-sided limits, justify why $\lim_{x\to2}\dfrac{x+3}{x-2}$ does not exist.

26. AdvancedEvaluate $\lim_{x\to0}\dfrac{\frac{1}{x+1}-1}{x}$ .

27. AdvancedUse the squeeze theorem to evaluate $\lim_{x\to0}x\sin(1/x)$ , and state the two bounding functions.

28. AdvancedEvaluate $\lim_{x\to9}\dfrac{x-9}{\sqrt{x}-3}$ .

29. AdvancedFind $a$ and $b$ so that $f(x)=ax+1$ for $x<3$ and $f(x)=x^2-b$ for $x\ge3$ is continuous at $x=3$ with $f(3)=7$ .

30. AdvancedFor $g(x)=\dfrac{x^2-x-6}{x^2-9}$ , classify the discontinuities at $x=3$ and $x=-3$ .