Limits and continuity - One Line Questions
1.
Consider the function f(x) = |x|/x for x != 0. What is lim (x->0+) f(x)? —
1
2.
Consider the function f(x) = |x|/x for x != 0. What is lim (x->0-) f(x)? —
-1
3.
Evaluate lim (x->pi) sin(x). —
0
4.
What is the value of lim (x->0) (sin(x)/x)? —
1
5.
Evaluate lim (x->2) (x^2 - 4)/(x - 2). —
4
6.
What is the value of lim (x->0) (e^x - 1)/x? —
1
7.
What is the limit of x^n as x approaches infinity, where n > 0? —
Infinity
8.
What is lim (x->0) cos(x)? —
1
9.
Consider the function f(x) = 1/x. What is lim (x->0) f(x)? —
Does not exist
10.
What is lim (x->0) (tan(x)/x)? —
1
11.
Consider f(x) = {x if x is rational, 0 if x is irrational}. What is lim (x->0) f(x)? —
0
12.
Evaluate lim (x->1) (x^3 - 1)/(x - 1). —
3
13.
What is lim (x->0) (1 - cos(x))/x? —
0
14.
Evaluate lim (x->e) ln(x). —
1
15.
Evaluate lim (x->infinity) (3x^2 + 2x - 1)/(x^2 + 5). —
3
16.
What is the limit of the sequence a_n = (1 + 1/n)^n as n approaches infinity? —
e
17.
For lim (x->a) f(x) to exist, the left-hand limit and right-hand limit must be equal. What if f(x) = x for x<0 and f(x) = x+1 for x>=0? What is lim (x->0) f(x)? —
Undefined
18.
What is lim (x->pi/2) cos(x)? —
0
19.
What is the value of lim (x->infinity) (sin(x)/x)? —
0
20.
What is the limit of the function f(x) = 2x + 3 as x approaches 5? —
13
21.
What is the left-hand limit of f(x) = x^2 at x=3? —
9
22.
What is the right-hand limit of f(x) = x^2 at x=3? —
9
23.
What is the value of lim (x->0) (sin(ax)/bx) where a and b are non-zero constants? —
a/b
24.
If lim (x->a) f(x) = L, what is lim (x->a) [c * f(x)], where 'c' is a constant? —
c * L
25.
If a function f(x) is continuous everywhere, it is also called: —
A continuous function
26.
If f(x) is continuous at x=c, then lim (x->c) f(x) = ? —
f(c)
27.
If lim (x->a) f(x) = L, what does this imply? —
As x gets arbitrarily close to 'a', f(x) gets arbitrarily close to L
28.
Which of the following is NOT a condition for a function f(x) to be continuous at a point 'c'? —
The function must be differentiable at 'c'.
29.
A function f(x) is continuous on an open interval (a, b) if it is continuous at every point in the interval. Which of the following functions is continuous on (-infinity, infinity)? —
f(x) = x^2 + 3x - 5
30.
Evaluate lim (x->infinity) (1/x). —
0
31.
What is the limit of f(x) = c (a constant) as x approaches infinity? —
c
32.
Which theorem states that if a function is continuous on a closed interval [a, b], then it attains its maximum and minimum values on that interval? —
Extreme Value Theorem
33.
If a function f(x) is continuous on a closed interval [a, b], what property does it possess according to the Intermediate Value Theorem? —
It takes on every value between f(a) and f(b).
34.
If lim (x->c) f(x) = L and L is a finite number, but f(c) is not defined, what type of discontinuity does f(x) have at x=c? —
Removable discontinuity
35.
If lim (x->a) f(x) = L and lim (x->a) g(x) = M, then what is lim (x->a) [f(x) * g(x)]? —
L * M
36.
If lim (x->a) f(x) = L and lim (x->a) g(x) = M (where M != 0), then what is lim (x->a) [f(x) / g(x)]? —
L / M
37.
If lim (x->a) f(x) = L and lim (x->a) g(x) = M, then what is lim (x->a) [f(x) + g(x)]? —
L + M
38.
If lim (x->a) f(x) = L, what is lim (x->a) [f(x)]^n? —
L^n
39.
What does it mean for a function to have an infinite discontinuity at x=c? —
Either lim (x->c+) f(x) or lim (x->c-) f(x) is infinity or -infinity
40.
What is the limit of a polynomial function P(x) as x approaches 'a'? —
P(a)
41.
What type of discontinuity occurs when lim (x->c) f(x) exists, but f(c) is not defined or f(c) is not equal to the limit? —
Removable discontinuity
42.
If lim (x->a) f(x) = L, what is lim (x->a) [sqrt(f(x))]? (Assume f(x) >= 0 near a and L >= 0) —
sqrt(L)
43.
For a limit to exist at a point 'c', what condition must be met? —
The left-hand limit and the right-hand limit must be equal.
44.
What is a 'jump discontinuity'? —
The left-hand limit and right-hand limit are finite but unequal.
45.
What is the fundamental concept of a limit in calculus? —
The behavior of a function as its input approaches a certain value.
46.
The limit of a constant function f(x) = k as x approaches any value 'a' is: —
k
47.
Is the function f(x) = x^2 continuous at x=3? —
Yes
48.
If lim (x->a) f(x) exists, and lim (x->a) g(x) exists, does lim (x->a) [f(x) / g(x)] necessarily exist? —
No, unless lim (x->a) g(x) != 0
49.
Consider the function f(x) = {x^2 if x < 0, x if x >= 0}. Is f(x) continuous at x=0? —
Yes
50.
Is the function f(x) = sin(1/x) continuous at x=0? —
No, because the limit as x approaches 0 does not exist (oscillating discontinuity).