Limit of a Function

LIMIT OF A FUNCTION

The essential idea of limit of a function is “nearness" to a point.

Roughly, the limit process involves examining the behaviour of the function ƒ near a point a. The nearness is restricted to an open interval of the form (a - δ, a + δ), where δ is a small positive number. This open interval is called a neighborhood of a.

Limiting behaviour occurs in a variety of practical problems.

For example,

Absolute zero, the temperature T at which all molecular activity ceases, can be approached but never actually attained.

Similarly engineering profiling the ideal specification of a new engine are really dealing with limiting behaviours.

To motivate mathematically, consider the function

Suppose we want to know what happens if x approaches 1.

We tabulate near 1 on the right side and left side of 1 in a neighborhood (1 − 0.1, 1 + 0.1) = (0.9, 1.1)

x approaches 1 from left →  ← x approaches 1 from right

We find when x is close to 1 from the left or from the right, f(x) is close to 2 

⸫ f(x) approaches 2 as x approaches 1 from either side. 

Generalising this we now give an informal definition of limit.

Definition 2.9 Limit

Let f be a function defined in a neighborhood of a, except possibly at the point a. If f(x) is arbitrarily close to l for all x sufficiently close to a from either side, then we say that f(x) approaches l as x approaches a and we write lim f(x) = l 

This is read as "the limit of f(x) as x approaches a is l"

Note:

We shall now give the precise definition of limit.

Definition 2.9(a) Precise definition of limit

Let f be a function defined in a neighbourhood of a, except possibly at the point a. Then f(x) = l means that for every ∈ > 0, there exists δ > 0 such that

Note 

In this definition the modulus inequalities can be rewritten in terms of interval as stated as below

means that for every ∈ > 0, there exists a δ > 0 such that

It is illustrated in the graph.

To evaluate limits the definition alone is not enough. The next theorem gives the basic properties of limit.

Theorem 2.1 

Note: If f(x) < g(x) ∀ x ∈ D, we cannot say but we can only conclude

Theorem 2.2 Sandwich theorem or the squeeze theorem

As a consequence of the basic limits, we have the following standard limits

Remark: Suppose exists, then need not exist.

WORKED EXAMPLES

Evaluate the following limits

Example 1 

Solution 

Example 2

Evaluate

Solution 

Given

Example 3 

Evaluate

Solution 

Given 

Example 4 

Solution

Example 5 

Evaluate

Solution 

Example 6 

Solution 

Example 7

Solution 

Given

Example 8 

Solution 

Given

Example 9 

Solution 

Example 10 

Evaluate

Solution 

Example 11 

Evaluate

Solution 

Example 12 

Evaluate

Solution 

Example 13 

Evaluate

Solution 

Example 14 

Solution 

Given

Since denominator → 0 as t → 0, we cannot use quotient rule of limits.

First we simplify by rationalizing the numerator.

Multiply numerator and denominator by

Example 15 

Solution 

Given

As x → 7, Denominator → 0

⸫ first simplify by rationalizing the Numerator,

Example 16 

Solution 

⸫ first simplify by rationalizing the Denominator,

Example 17 

Solution 

Given

As x → 0, Denominator → 0

⸫ first simplify by rationalizing the Numerator,

1. One-Sided Limits

Sometimes the limit of a function may not exist but only one side limit may exist i.e., either left side limit or right side limit may exist.

Definition 2.10 Left-hand limit or left limit

If the values f(x) of a function ƒ approach arbitrarily close to l for all x sufficiently close to a from the left of a (i.e, x < a), then the left-hand limit exists and is written as

Similarly, if the values f(x) of ƒ approach arbitrarily close to l for all x sufficiently close to a from the right of a (i.e, x > a) then the right-hand limit exists and is written as

For Example If f(x) =

Solution

The function is not defined at x = 0 but defined in the neighborhood of 0.

⸫ at x = 0, left-hand limit = -1 and right-hand limit = 1

The next theorem connects limit and one-sided limit

Theorem 2.3 

Note:

By the theorem, in the above example not exist.

Because

WORKED EXAMPLES

Example 1 

If f(x) =

then determine whether

Solution 

Since on the left side and right side of 4, ƒ is defined differently, we find one- sided limits.

Left limit: 8 − 2.4 = 0

Right limit:

Example 2 

Evaluate

Solution 

Example 3 

Find the limit if it exists? If the limit does not exist explain why?

Solution 

Given limit is the left limit

Example 4 

Solution 

Example 5 

(i) Evaluate the following

(ii) Sketch the graph of ƒ

Solution 

(ii) If x < 1, then the graph is y = x, which is a straight line bisecting the angle between the positive x and y axis.

If x = 1, then the graph is a point (1, 3).

If 1 < x ≤ 2, then the graph is y = 2 - x2 ⇒ x2 = 2 – y = - (y - 2)

which is a parabola (of the form x2 = -4ay), with vertex (0, 2), and the axis is x = 0

i.e., the y-axis and downward

If x > 2, then the graph is y = x - 3 which is a straight line

Put x = 0, then y = -3 and put y = 0, then x = 3

⸫ two points on it are (0, --3), (3, 0)

We shall now draw the graph

Example 6 

Show that

Solution 

Example 7

If f(x) = [x] is the greatest integer function, then find the limits if it exists.

Solution 

Given f(x) = [x]

2. Extended Real Number System

With the real number set ℝ we add two symbols -∞ and +∞ or ∞ and get a larger set ℝ* = ℝ∪{-∞, ∞}

ℝ* is called the extended real number system subject to the following rules

(i) −∞ < x ∞ ∀ x ∈ ℝ i.e., any real number lies between these symbols.

We also write ℝ = the open interval (-∞, ∞)

(ii) For any real number x, x + ∞ = ∞, x - ∞ = - ∞

(iii) ∞ + ∞ = ∞, -∞ + (-∞) = -∞

(iv) For any real number,

(v) If x > 0, x·∞ = ∞, x⋅(-∞) = -∞ 

If x < 0, x.∞ = -∞, x⋅(-∞) = ∞

(vi) ∞.∞ = ∞, -∞·∞ = -∞ = ∞·(-∞)

(-∞)(-∞) = ∞

However, are indeterminates

In our discussions whenever -∞ or ∞ is involved then we are discussing in ℝ*

3. Infinite Limits

Consider the function

In a neighborhood of 0, when we approach 0 from the right side 1/x is increasing indefinitely i.e.,

When we approach 0 from the left (i.e., through negative values) 1/x decreases indefinitely i.e.,

We describe these limiting behaviours by writing and

The graph of ⇒ xy = 1, which is the rectangular hyperbola

Definition 2.11 Infinite limits

Let ƒ be a function defined in a neighbourhood of a, except possible at a.

Definition 2.12 Vertical Asymptote

The line x = a is called a vertical asymptote to the curve y = f(x) if

For the curve y = 1/x, the line x = 0 or y-axis is a vertical asymptote because

The word asymptote comes from the Greek word 'asymptotos' which means non intersecting.

Definition 2.13 Horizontal asymptote

A line y = b is a horizontal asymptote of the curve y = f(x) if either

For the curve y = 1/x, the line y = 0 or x-axis is a horizontal asymptote because

Note

In the evaluation of limits of rational function and irrational functions involving polynomial as x → ± ∞, we manipulate the function so that the powers of x are transformed to powers of 1/x· This is done by pulling out the highest power of x in the numerator and denominator.

WORKED EXAMPLES

Evaluate the following limits

Example 1 

Solution

Second Method: Another method is by the transformation of the variable x

Example 2 

Solution

Example 3

Solution

Example 4 

Solution

Second Method:

Example 5 

Prove that

Solution

4. Limits with Trigonometric Functions

Theorem 2.4

Proof:

Consider a circle with centre O and radius r.

Let θ be a small angle in radian and let AOP =

Draw PN ⊥ OA and the tangent at A meet OP produced at Q. 

From figure, we find that

Area of ΔAOP < Area of sector AOP < Area of ΔAOQ

Replacing θ by - θ, the in equality holds.

So, the inequality holds for < θ < 0 also.

As θ → 0, P approaches along the arc of the circle to A and ON→OA = r

In this formula θ is radian.

WORKED EXAMPLES

Evaluate the following limits

Example 1 

Solution

Example 2 

Solution

Example 3 

Solution

Example 4 

Solution

Example 5 

Solution

In limits with trigonometric functions if the variable x tends to a nonzero number, say then we convert it to tend to zero by putting

Example 6 

Solution

Example 7 

Solution 

Example 8 

Solution 

Example 9 

Solution 

Example 10 

Solution 

Example 11 

Solution 

Example 12 

Solution 

5. Limits with Exponential and Logarithmic Functions Standard Limits

WORKED EXAMPLES

Evaluate the following limits

Example 13 

Solution 

Example 14 

Solution 

Example 15 

Solution 

Example 16 

Solution 

Example 17 

Solution 

Example 18 

Solution 

EXERCISE 2.3

Evaluate the following limits

ANSWERS TO EXERCISE 2.3