Limits & Continuity — Practice
Where a function is heading matters more than whether it ever arrives.
What these questions 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.
Problem 1
Find limx → 3(2x+1).
Worked answer
7
2(3)+1 = 7
Problem 2
Find limx → 2(x2-5).
Worked answer
-1
4-5 = -1
Problem 3
Find limx → 0(3x2+4x+9).
Worked answer
9
Only the constant survives
Problem 4
Find limx → 4x+2x-1.
Worked answer
2
63 = 2
Problem 5
What does limx → a f(x) = L tell you about f(a)?
- Nothing
- That f(a)=L
- That f(a) is undefined
- That f(a) ≠ L
Worked answer
Nothing
The single point x=a is deliberately ignored
Problem 6
Find limx → 2x2-4x-2.
Worked answer
4
(x-2)(x+2)x-2 = x+2
At x=2 that is 4
Problem 7
Find limx → 3x2-9x-3.
Worked answer
6
(x-3)(x+3)x-3 = x+3
At x=3 that is 6
Problem 8
Find limx → 1x2-1x-1.
Worked answer
2
x+1 away from x=1
At x=1 that is 2
Problem 9
Find limx → -2x2+5x+6x+2.
Worked answer
1
(x+2)(x+3)x+2 = x+3
At x=-2 that is 1
Problem 10
Substituting gives 00. What does that mean?
- More work is needed
- The limit is 0
- The limit is 1
- The limit does not exist
Worked answer
More work is needed
It is an indeterminate form
Top and bottom share a factor that has not been cancelled
Problem 11
For f(x)=2x when x<1 and f(x)=x+5 when x ≥ 1, find the limit as x approaches 1 from the left.
Worked answer
2
2(1) = 2
Problem 12
For the same function, find the limit as x approaches 1 from the right.
Worked answer
6
1+5 = 6
Problem 13
For that same function, does limx → 1 f(x) exist?
- No — the two sides disagree
- Yes, it is 2
- Yes, it is 6
- Yes, it is 4
Worked answer
No — the two sides disagree
2 from the left and 6 from the right
They must agree for the limit to exist
Problem 14
A function has a hole at x=2 but heads for 5 from both sides. Which is true?
- The limit is 5 but it is not continuous there
- It is continuous
- The limit does not exist
- The limit is 0
Worked answer
The limit is 5 but it is not continuous there
The limit exists; the value does not
Problem 15
Find limx → 0x2+3xx.
Worked answer
3
x(x+3)x = x+3
At x=0 that is 3
Problem 16
Find limx → 4x-4√x-2.
Worked answer
4
Bottom becomes x-4
So the expression is √x + 2, which is 4 at x=4