Limits

Question 1

What is the limit of a function?

Answer 1

Given a function f(x) defined at all values in an open interval with the variable ‘a’; if the value of x is made to approach a (where x ≠ a) and the resulting value of the function tends towards a fixed point L (where L is a real number), then L is said to be the limit of the function.

Question 2

What is the symbol for the limit of a function?

Answer 2

Lim_x→af(x) = L

Question 3

What is the left-handed limit of a function

Answer 3

Let the function f(x) be defined at all values in an open interval of (b, a) where b is a constant; if value of the function tends to a real number L as x approaches ‘a’ (such that x < a) then L is said to be the left-handed limit of the function.

Lim(x→a-)f(x) = L

Question 4

What is the right-handed limit of a function

Answer 4

Let the function f(x) be defined at all values in an open interval of (a, c) where c is a constant; if value of the function tends to a real number L as x approaches ‘a’ (such that x > a) then L is said to (be the right-handed limit of the function) be the limit of f(x) as x approaches ‘a’ from the right.

Lim(x→a⁺)f(x) = L

Question 5

What is the relationship between one-sided limits and two-sided limits?

Answer 5

Lim(x→a)f(x) = L if and only if Lim(x→a+)f(x) = L , AND Lim(x→a-)f(x) = L

Question 6

What are the properties for limits of a function of the nature f(x) = 1/(x - a)ⁿ ?

Answer 6

• If n is a positive odd number, then:
  lim(x→a-) 1/(x - a)ⁿ = –∞
  and
  lim(x→a+) 1/(x - a)ⁿ = +∞
• If n is a positive even integer then:
  lim(x→a) 1/(x - a)ⁿ = +∞

Question 7

What is a two-sided infinite limit?

Answer 7

Let a function f(x) be defined at all values in an open interval containing the variable ‘a’. If x is made to approach ‘a’ (such that x ≠ a) and the value of the function either increases or decreases boundlessly, then the limit of the function as x tends to a is infinity; positive infinity if the function increases boundlessly, and negative infinity if it decreases.

Question 8

What is an infinite limit from the left?

Answer 8

Let a function f(x) be defined at all values in an open interval (b, a). If x is made to approach ‘a’ (such that x < a) and the value of the function either increases or decreases boundlessly, then the function is said to have a limit of positive or negative infinity as x tends to ‘a’ from the left; positive infinity from the left if the function increases boundlessly, and negative infinity if it decreases.

Question 9

What is an infinite limit from the right?

Answer 9

Let a function f(x) be defined at all values in an open interval (a, c). If x is made to approach ‘a’ (such that x > a) and the value of the function either increases or decreases boundlessly, then the function is said to have a limit of positive or negative infinity as x tends to ‘a’ from the right; positive infinity from the right if the function increases boundlessly, and negative infinity if it decreases.

Question 10

What is a vertical asymptote of a function f(x)?

Answer 10

A vertical asymptote is said to exist on the line x = a if the following conditions are met:

lim(x→a+)f(x) = +∞ or -∞ and lim(x→a-)f(x) = +∞ or -∞

Or,

lim(x→a)f(x) = +∞ or -∞

Question 11

What are the Basic Limit laws?

Answer 11

Lim(x→a) x = a

Lim(x→a) c = c (where c is a constant)

Question 12

What is the sums law of limits?

Answer 12

lim(x→a)(f(x) + g(x)) =
lim(x→a)f(x) + lim(x→a)g(x)

Question 13

What is the difference law of limits?

Answer 13

lim(x→a) (f(x) - g(x)) = lim(x→a) f(x) - lim(x→a) g(x)

Question 14

What is the constant multiplicity law of limits?

Answer 14

lim(x→a) c•f(x) = c•lim(x→a)f(x)

Question 15

What is the product law of limits?

Answer 15

lim(x→a) (f(x)•g(x)) = lim(x→a)f(x) • lim(x→a)g(x)

Question 16

What is the quotient law of limits?

Answer 16

lim(x→a) f(x)/g(x) = lim(x→a)f(x) ÷ lim(x→a)g(x)

As long as g(x) ≠ 0

Question 17

What is lim(θ→0)sinθ?

Answer 17

lim(θ→0)sinθ = 0

Question 18

What is lim(θ→0)cosθ ?

Answer 18

lim(θ→0)cosθ = 1

Question 19

What is lim(θ→0)sinθ/θ ?

Answer 19

lim(θ→0)sinθ/θ = 1

Question 20

What is squeeze theorem?

Answer 20

Let the functions f(x), g(x), and h(x) be defined for all x ≠ a over an open interval containing a.
If f(x) ≤ g(x) ≤ h(x),
and lim(x→a)f(x) = L = lim(x→a)h(x)
Where L is a real integer, then
lim(x→a)g(x) = L