Limits and Continuity Problems with Solutions PDF Summary
Dear readers, here we are offering limits and continuity problems with solutions pdf to all of you. Santana formula is one of the most important topics of class 12th maths because it is widely used in calculus. Along with this, inverse trigonometry formulas and only trigonometry functions are also used more. Well, Continuity is based on Inverse Trigo and Function. A function is said to be continuous if it is continuous at every point in its domain. A function is determined to be continuous on an interval, or a subset of its domain, if and only if it is continuous at every point in its domain.
Addition, subtraction, and multiplication of continuous functions with the same domain are also continuous except at a point in which the denominator is equal to zero. Continuity can also be defined with respect to limits by saying that f(x) is continuous at x₀ of its domain if and only if, for values of x in its domain, Limit And Continuity is the most important concept which is also called Limit. In which the complete study is done about the geometric graph. The formula provided here is only to improve maths study and complete a smart preparation plan who are aspirants to clear 12th class
Limits and Continuity Problems with Solutions PDF 2023
Many functions have the property that they can trace their graphs with a pencil without lifting the pencil from the paper’s surface. These types of functions are called continuous. Intuitively, a function is continuous at a particular point if there is no break in its graph at that point. A precise definition of continuity of a real function is provided generally in a calculus’s introductory course in terms of a 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), when x approaches the point “a”, is equal to the value of f(x) at “a”, i.e., f(a). Second, the function (as a whole) is continuous, if it is continuous at every point in its domain.
Mathematically, continuity can be defined as given below:
A function is said to be continuous at a particular point if the following three conditions are satisfied.
As mentioned before, a function is said to be continuous if you can trace its graph without lifting the pen from the paper. But a function is said to be discontinuous when it has any gap in between.
Below figure shows the graph of a continuous function.
Limit Definition
A limit of a function is a number that a function reaches as the independent variable of the function reaches a given value. The value (say a) to which the function f(x) gets close arbitrarily as the value of the independent variable x becomes close arbitrarily to a given value “A” symbolized as f(x) = A.
Points to remember:
- 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 known 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 known as the right-hand limit of f(x) at a.
- If the right-hand and left-hand limits coincide, we say the common value as the limit of f(x) at x = a and denote it by limx→a f(x).
One-Sided Limit
The limit is based completely on the values of a function taken at x -value that is slightly greater or less than a particular value.
Properties of Limit – Limits and Continuity Problems with Solutions PDF
- The limit of a function is represented as f(x) reaches L as x tends to limit a, such that; limx→af(x) = L
- The limit of the sum of two functions is equal to the sum of their limits, such that: limx→a [f(x) + g(x)] = limx→a f(x) + limx→a g(x)
- The limit of any constant function is a constant term, such that, limx→a C = C
- The limit of product of the constant and function is equal to the product of constant and the limit of the function, such that: limx→a m f(x) = m limx→a f(x)
- Quotient Rule: limx→a[f(x)/g(x)] = limx→af(x)/limx→ag(x); if limx→ag(x) ≠ 0
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