Limits & Continuity — Drills
Limits & Continuity has 5 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Factor — A number that divides evenly into another.
- Function — A rule pairing each input with exactly one output.
- Polynomial — A sum of terms, each a number times a whole-number power of the variable.
- Limit — The value a function approaches as the input approaches some number.
- One-sided limit — The value approached from the left, or from the right, alone.
- Continuous — A function you could draw without lifting the pen — no holes, jumps or blow-ups.
- Indeterminate form — 00, which means the limit needs more work — not that it fails to exist.
Substitution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find limx → -1(5x2 + 0x - 4).
Worked answer
1
5(1) + 0(-1) - 4
= 1
Problem 2
Find limx → -2(1x2 - 3x + 6).
Worked answer
16
1(4) - 3(-2) + 6
= 16
Factoring
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find limx → 2x2-4x-2.
Worked answer
4
(x-2)(x+2)x-2 = x+2
At x=2 that is 4
Problem 2
Find limx → 4x2-16x-4.
Worked answer
8
(x-4)(x+4)x-4 = x+4
At x=4 that is 8
Factor2
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find limx → -7x2+12x+35x+7.
Worked answer
-2
(x+7)(x+5)x+7 = x+5
At x=-7 that is -2
Problem 2
Find limx → -1x2+5x+4x+1.
Worked answer
3
(x+1)(x+4)x+1 = x+4
At x=-1 that is 3
Onesided
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For f(x)=4x when x<4, and f(x)=x+5 when x ≥ 4, find the limit as x approaches 4 from the left.
Worked answer
16
4(4) = 16
Problem 2
For f(x)=5x when x<2, and f(x)=x+4 when x ≥ 2, find the limit as x approaches 2 from the left.
Worked answer
10
5(2) = 10
Cancel
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find limx → 0x2+2xx.
Worked answer
2
x(x+2)x = x+2
At x=0 that is 2
Problem 2
Find limx → 0x2+4xx.
Worked answer
4
x(x+4)x = x+4
At x=0 that is 4