Limit Definition of Derivative and Its Relationship to Continuity

Definition of Continuity

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Many functions have the virtue of being able to trace their graphs with a pencil without removing the pencil off the paper. These are called Continuous functions, a function is continuous at a given point if its graph does not break at that point. In general, a calculus introductory course will provide a clear description of continuity of a real function in terms of the limit's idea. First, a function f with variable x is continuous at the point "a" on the real line if the limit of f(x), as x approaches "a," is equal to the value of f(x) at "a," i.e., f(a).

Read Also: Integration by Partial Fractions

Continuity can be described mathematically as follows:

If the following three conditions are met, a function is said to be continuous at a given point.

1. f(a) is defined
2. lim _x→a f(x) exists
3. lim_x→a f(x)=lim _x→a f(x)=f(a)

When a graph can be traced without lifting the pen from the sheet, the function is said to be a continuous function. A function, on the other hand, is said to be discontinuous if it contains any gaps in between.

Read More: Conditional Probability

The graph of a continuous function is shown below:

The graph of a continuous function

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Types of Discontinuity

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• Infinite discontinuity: A branch of discontinuity with a vertical asymptote at x = a and f(a) is not defined. Asymptotic Discontinuity is another name for this. A function can't be connected if it has values on both sides of an asymptote, therefore it's discontinuous at the asymptote.
• Jump discontinuity: A branch of discontinuity in which limx→a+f(x)≠limx→a−f(x), but of the both limits are finite. This is also known as simple discontinuity or continuity of the first kind.
• Positive Discontinuity: A branch of discontinuity in which a function has a predefined two-sided limit at x = a, but f(x) is either undefined or not equal to the limit at a. A removable discontinuity is another name for this.

These can be represented graphically as:

Types of Discontinuity

Read Further: Integration by Parts

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Definition of Limit

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A function's limit is a number that a function reaches when its independent variable reaches a certain value. The value (say a) to which the function f(x) approaches arbitrarily as the independent variable x approaches arbitrarily a given value "A" denoted as f(x) = A.

If limx→a- f(x) is the expected value of f at x = a given the values of 'f' near x to the left of a. This value is referred to as the left-hand limit of 'f' at a.

If limx→a+ f(x) is the expected value of f at x = a given the values of 'f' near x to the right of a. This value is referred to as the right-hand limit of f(x) at a.

If the right-hand and left-hand limits coincide, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).

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One-Sided Limit

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The limit that is entirely determined by the values of a function for an x-value that is slightly higher or less than a given value. A two-sided limit lim x→af(x)lim x→af(x) takes the values of x into consideration that are both larger than and smaller than a. A one-sided limit from the left limx→a−f(x)limx→a−f(x) or from the right limx→a−f(x)limx→a−f(x) takes only values of x that is smaller or bigger than a respectively.

Read More: Methods of Integration

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Points to Remember

• When a graph can be traced without lifting the pen from the sheet, the function is said to be a continuous function. A function, on the other hand, is said to be discontinuous if it contains any gaps in between.
• If limx→k− f(x)limx→k−f(x) is the expected value of f at x = k stated the value of 'f' is close by x to the left side of k. The left-hand limit of 'f' at k is used to determine this value.
• If limx→k+ f(x) is the average value of f at x = k, then the values of 'f' on the right side of k are lim x→k+f(x). The right-hand limit of f(x) at k is used to determine this value.
• If the right-hand and left-hand limits meet, the common value is called the limit of f(x) at x = k and is denoted by limx→kf(x).
• The limit of any constant function will always be a constant term limx→yC= C.

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Sample Questions

Ques 1. Find the value of lim _x→3 [x(x+2)]. (1 marks)

Ans. lim_x→3 [x(x+2)] = 3(3+2) = 3 x 5 = 15

Ques 2. Check the continuity of the function f given by f (x) = 2x + 3 at x = 1. (NCERT) (2 marks)

Ans. As we know that the function is defined at the given point x = 1 and its value is 5. Finding limit of the function at x = 1.

lim_x→1 f(x) = lim_x→1 f(2x+3) = 2(1)+3 = 5

lim_x→1 f(x) = 5 = f(1)

Therefore, f is continuous at x = 1.

Ques 3. Check the points where the constant function f (x) = k is continuous. (NCERT) (2 marks)

Ans. Let c be any real number.

Then lim_x→c f(x) = lim_x→c k = k

Since f(c) = k = lim_x→c f (x) for any real number c, Thus in this case, the function f is continuous at every real number.

Ques 4. Compute: limx→−2 (3x2+5x−9) (2 marks)

Ans. Using property 2 to divide the limit in 3 different limits. After that, using property 1 to bring the constants out of the first two. We get,

lim_x→−2(3x2+5x−9)=lim_x→−2 (3x2)+lim_x→−2(5x)−lim_x→−2(9)=3(−2)2+5(−2)−(9)=3(−2)2+5(−2)−(9) = 12 – 10 – 9 = -7

Ques 5. Discuss the continuity of sine function. (2 marks)

Ans. lim _x→0 sinx=0

It is intuitively clear from the graph of sin x near 0.

Let's observe that f (x) = sin x is defined for every real number. Let c be a real number. Put x = c + h. If x → c we know that h → 0.

Therefore,

lim_x→c f(x) = lim→x sinx = lim_h→0 sin(c+h) = lim _h→0 [sinc cosh + cosc sinh] = lim _h→0 [sinc cosh] + lim _h→0 [cosc + sinh] = sinc c + 0 = sin c = f(c)

Thus, lim_x→c f(x)=f(c) and f is a continuous function.

Ques 6. Show that the function defined by f(x) = sin (x2) is a continuous function. (2 marks)

Ans. The function is defined for every real number. The function f may be thought of as a composition g o h of the two functions g and h, where g (x) = sin x and h(x) = x2 . By Theorem 2, we can deduce that f is a continuous function because both g and h are continuous functions.

Ques 7. Show that the function f defined by f (x) = |1 – x + | x| |, where x is any real number, is a continuous function. (2 marks)

Ans. Let us define g by g (x) = 1 – x + | x| and h by h (x) = | x| for all real x.

Then (h o g) (x) = h (g (x)) = h (1– x + | x|) = | 1– x + | x| | = f (x)

Ques 8. Examine the continuity of the function f(x) = 2x2 – 1 at x = 3. (2 marks)

Ans. The given function is f(x) = 2x2 – 1

At x=3, f(3)=2(3×3)-1 = 17

lim_x→3 f(x)= lim_x→3 2x2 – 1= 2(3×3)-1 = 17

lim_x→3 f(x) = f(3)

therefore, f is continuous at x=3.

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