LIMITS OF TRIGNOMETRIC FUNCTIONS
Sine Function
The function f(x) = sin(x) is a continuous function over its entire domain, with its domain consisting of all the real numbers. The range of this function is [-1, 1] as can be seen in the graph below:
So, if the limit of sine function is calculated at any given real number it’s always defined and lies between [-1, 1].
f(x) = sin(x)
limx→a f(x) = limx→a sin(x)
= sin(a), where a is a real number
Cosine Function
The function f(x) = cos(x) is a continuous function over its entire domain, with its domain consisting of all the real numbers.
The range of this function is [-1, 1] as can be seen in the graph below:
So, if the limit of the cosine function is calculated at any given real number it’s always defined and lies between [-1, 1].
f(x) = cos(x)
limx→a f(x) = limx→a cos(x)
= cos(a), where a is a real number
Tangent Function
The function f(x) = tan(x) is defined at all real numbers except the values where cos(x) is equal to 0, that is, the values π/2 + πn for all integers n. Thus, its domain is all real numbers except π/2 + πn, n € Z.
The range of this function is (-∞, +c).
So, if the limit of the tangent function is calculated in its domain it always defined and lies between(-∞, +∞).
f(x) = tan(x)
limx→a f(x) = limx→a tan(x)
= tan(a), where a belongs to real no. except π/2 + πn, n € Z
The function f(x) = cosec(x) is defined at all real numbers except the values where sin(x) is equal to 0, that is, the values πn for all integers n. Thus, its domain is all real numbers except πn, n € Z.
The range of this function is (-∞,-1] U [1,+∞).
So, if the limit of the cosine function is calculated in its domain it always defined and lies between its range.
f(x) = cosec(x)
limx→a f(x) = limx→a cosec(x)
= cosec(a), where a belongs to real no. except for πn , n € Z
Secant Function
The function f(x) = sec(x) is defined at all real numbers except the values where cos(x) is equal to 0, that is, the values π/2 + πn for all integers n. Thus, its domain is all real numbers except π/2 + πn, n € Z.
The range of this function is (-∞, -1] U [1, +∞)
So, if the limit of sec function is calculated in its domain it always defined and lies between its range.
f(x) = sec(x)
limx→a f(x) = limx→a sec(x)
= sec(a), where a belongs to real no. except for π/2 + πn, n € Z
Cot Function
The function f(x) = cot(x) is defined at all real numbers except the values where tan(x) is equal to 0, that is, the values πn for all integers n. Thus, its domain is all real numbers except πn, n € Z.
The range of this function is (-∞, +∞)
So, if the limit of the cot function is calculated in its domain it always defined and lies between its range.
f(x) = cot(x)
limx→a f(x) = limx→a cot(x)
= cot(a), where a belongs to real no. except for πn, n € Z
Limits of Trigonometric Functions
The following facts (stated as theorems) about functions in general come in handy in calculating limits of some trigonometric functions.
Theorem Let f and g be two real valued functions with the same domain such that f(x) ≤ g( x) for all x in the domain of definition, For some a, if both )and limₓ→ₐf(x) and limₓ→ₐ g(x) exist, then limₓ→ₐf(x) and limₓ→ₐ g(x). This is illustrated in
SANDWICH THEOREM
Theorem (Sandwich Theorem)
Let f g and h be real functions such that f(x)≤g(x)≤h(x) for all x in the common domain of definition. For some real number a, if limₓ→ₐfx) = l = limₓ→ₐh(x), then limₓ→ₐg(x) = 1.
