Behaviour of functions

Question 1

f(x) tends to infinity as x tends to infinity

Answer 1

Suppose that f is a real function, f(x) tends to infinity as x tends to infinity, if given any A in R+, there exists K in R s.t.
f(x) > A whenever x> K.

Question 2

f(x) tends to minus infinity as x tends to infinity

Answer 2

f(x) tends to minus infinity as x tends to infinity, if given any A in R+ there exists K in R such that f(x) < -A whenever x>K

Question 3

f(x) tends to l as x tends to infinity

Answer 3

f(x) tends to l as x tends to infinity, if given an e in R+, there exists K in R such that |f(x) - l| < e whenever x>K.

Question 4

Limit of 1/f(x) as x tends to infinity

Answer 4

Suppose that f(x) tends to infinity as x tends to infinity then
1/f(x) = 0 as x tends to infinity

Question 5

f(x) tends to infinity as x tends to minus infinity

Answer 5

if given any A in R+, there exists K in R s.t.

f(x) > A whenever x < K

Question 6

f(x) tends to minus infinity as x tends to minus infinity

Answer 6

if given any A in R+, there exists K in R s.t.

f(x) < - A whenever x < K.

Question 7

f(x) tends to l as x tends to minus infinity

Answer 7

if given any e in R+, there exists K in R such that

| f(x) - l | < e whenever x < K.

Question 8

Algebra of limits

Answer 8

Suppose that lim (x tends to infinity) of f(x) = l, and lim (x tends to infinity) of g(x) = m where l and m e R. Then

(1) lim(f(x) + g(x)) = l + m
(2) lim (f(x) - g(x) ) = l - m
(3) lim (f(x)g(x) ) = lm
(4) if m=/ 0, the lim (f(x)/g(x)) = l/m
(Also hold on (a-R, a) U (a, a+R), when x tends to a)

Question 9

Sandwich theorem

Answer 9

Let f, g and h be real functions defined on (R, inf).
Suppose f(x) =< g(x) =< h(x) for all x e (R, inf) and lim (f(x)) =l and lim (g(x)) = l where l e R. Then,
lim (g(x)) = l
(Also hold on (a-R, a) U (a, a+R), when x tends to a)

Question 10

Right hand limit

Answer 10

Suppose f is defined on ( a, a+R) . f(x) tends to l as x tends to a from above if given any e in R+, there exists d in R+ such that
| f(x)-l | < e whenever a < x< a+d

Question 11

Left hand limit

Answer 11

Suppose that f is defined on (a - R, a). f(x) tends to l as x tends to a from the left if given any e in R+ there exists d in R+ such that
| f(x)-l | < e whenever a-d =< x =< a

Question 12

e-d definition of a limit at a point

Answer 12

Suppose f is defined on (a-R, a) U (a, a+R). f(x) tends to l as x tends to a if given any e in R+, there exists d in R+ such that
| f(x) -l | < e whenever 0 < |x-a| < d

Question 13

Punctured neighbourhood of a

Answer 13

{x e R: 0< |x-a| < d}

Question 14

Connection between sequences and functions

Answer 14

Suppose that f is a real function defined on (a-R, a) U (a, a+R).
Then lim, as x tends to a, of f(x) is l,
iff lim, as n tends to infinity f(an) = l for all sequences (an) s.t. lim, as n tends to infinity, of an =a and an is not equal to zero, for all n e N.

Question 15

Fundamental trigonometric limits

Answer 15

Lim, as x tends to 0, of cos(x) =1

Lim, as x tends to 0, of sin(x)/x =1

Question 16

Changing variables in a limit

Answer 16

Let a, b, c e R and b=/0. Suppose f is a real function whose domain includes at least a punctured neighbourhood of ba and a+c.
(1) If lim, as x tends to ba, of f(x) = l for some l e R, then the l
lim, as x tends to a, of f(bx) =l.
(2) If lim, as x tends to a+c, of f(x) =m, for some m e R, then lim, as x tends to a, of f(x+c) =m.

Question 17

Standard limits

Answer 17

(1) lim, as x tends to a, of x^n = a^n
(2) lim, as x tends to a, of f(bx+c) = lim, as x tends to ab+n, of f(x)
(3) lim, as x tends to 0, of sinx = 0
(4) lim, as x tends to 0 cos(x) =1
(5) lim, a x tends to 0, sin(x)/x =1