Engineering Mathematics | Gate Exam | Calculus

Calculus

⚫Introduction of Limits
⚪ Consider a function f(x). Now, if x approaches a value c, and if for any number a > 0, we find a number b > 0 such that |f(x) − l | < a whenever 0 < |x − c| < b, then l is called the limit of function f(x).
⚪ Limits are used to define continuity, derivatives and integrals.
⚪ Limits are denoted as
⚪ Left-Hand and Right-Hand Limits
If the values of a function f(x) at x = c can be made as close as desired to the number L₁ at points closed to c and on the left of c, then L₁ is called left-hand limit.

If the values of a function f(x) at x = c can be made as close as desired to the number L₂ at points on the right of c and close to c, then L₂ is called right-hand limit.

⚪ Properties of Limits
Some of the important uses formula of limits are:

⚪ L's Hospital Rule

Question based on L's Hospital Rule (Gate 2017) :
The value of :
Solution :

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Previous Year Question

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⚫ Continuity and Discontinuity
⚪ A function f(x) at any point x = c is continuous if ⚪ A function f(x) is continuous on a closed interval [a, b] if
⚪ If the conditions for continuity are not satisfied for a function f(x) for a point or an interval, then the function is said to be discontinuous.

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⚫ DIFFERENTIABILITY
⚪ Consider a real-valued function f(x) defined on an open interval (a, b). The function is said to be differentiable for x = a, if
⚪ A function is always continuous at a point if the function is differentiable at the same point. However, the converse is not always true
⚪ For a function being differentiated , we need to find right side differentiated function and left side differentiated function , and both must be equal.

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⚫ Rolle theorem : Consider a real-valued function defined in the closed interval [a, b], such that 1. It is continuous on the closed interval [a, b].
2. It is differentiable on the open interval (a, b).
3. f(a) = f(b).
Then, according to Rolle’s theorem, there exists a real number c ∈ (a,b) such that f'(c) = 0. ⚫ Lagrange’s Mean Value Theorem : Consider a function f(x) defined in the closed interval [a, b], such that 1. It is continuous on the closed interval [a, b].
2. It is differentiable on the closed interval (a, b).
Then, according to Lagrange’s mean value theorem, there exists a real number c ∈ (a,b), such that