# 11 Lecture

## Limits (Rigorous Approach)

Calculus is a branch of mathematics that deals with the study of change, and it is essential for understanding a wide range of phenomena in science, engineering, and economics.

## Important Mcq's Midterm & Finalterm Prepration Past papers included

1. What is the limit of f(x) as x approaches 2 if f(x) = x^2 - 3x + 2? A. 1 B. 2 C. 3 D. 4 Answer: D. 4

2. What is the limit of g(x) as x approaches 0 if g(x) = sin(x)/x? A. 0 B. 1 C. -1 D. Does not exist Answer: B. 1

3. What is the limit of h(x) as x approaches infinity if h(x) = 5/x? A. 0 B. 5 C. infinity D. Does not exist Answer: A. 0

4. What is the limit of j(x) as x approaches 1 if j(x) = (x - 1)/(x^2 - 1)? A. 0 B. 1 C. -1 D. Does not exist Answer: B. 1

5. What is the limit of k(x) as x approaches infinity if k(x) = (3x - 2)/(4x + 1)? A. 3/4 B. 2/3 C. 3/1 D. Does not exist Answer: A. 3/4

6. What is the limit of f(x) as x approaches 0 if f(x) = (2x + 1)/(x - 3)? A. 1/3 B. 2/3 C. -1/3 D. Does not exist Answer: D. Does not exist

7. What is the limit of g(x) as x approaches 2 if g(x) = (x^2 - 4)/(x - 2)? A. 0 B. 1 C. 2 D. Does not exist Answer: C. 2

8. What is the limit of h(x) as x approaches 3 if h(x) = sqrt(x - 3)? A. 0 B. 1 C. 3 D. Does not exist Answer: D. Does not exist

9. What is the limit of j(x) as x approaches infinity if j(x) = e^(-2x)? A. 0 B. 1 C. -1 D. Does not exist Answer: A. 0

10. What is the limit of k(x) as x approaches 1 if k(x) = (x - 1)^2/|x - 1|? A. 0 B. 1 C. Does not exist D. infinity Answer: C. Does not exist

## Subjective Short Notes Midterm & Finalterm Prepration Past papers included

1. What is the definition of a limit in calculus? Answer: The limit of a function f(x) as x approaches a is the value that f(x) approaches as x gets closer and closer to a.

2. What is the difference between a one-sided limit and a two-sided limit? Answer: A one-sided limit only considers the behavior of the function from one side of a point, while a two-sided limit considers the behavior of the function from both sides of the point.

3. What is an indeterminate form in calculus? Answer: An indeterminate form is a mathematical expression that is not well-defined, such as 0/0 or infinity/infinity.

4. What is L'Hopital's rule, and when is it used to evaluate limits? Answer: L'Hopital's rule is a method for evaluating limits that involve taking the derivative of the numerator and denominator separately and then evaluating the limit again. It is used when we have an indeterminate form of the type 0/0 or infinity/infinity.

5. What is the Squeeze theorem, and when is it used to evaluate limits? Answer: The Squeeze theorem is a method for evaluating limits that involve bounding a function between two other functions, and if the limits of the two bounding functions are equal, then the limit of the bounded function is also equal to that limit.

6. What is the meaning of a limit that equals infinity? Answer: If a limit equals infinity, it means that the function grows without bounds as x approaches the limiting value.

7. What is the meaning of a limit that equals negative infinity? Answer: If a limit equals negative infinity, it means that the function decreases without bound as x approaches the limiting value.

8. What is the difference between a removable and non-removable discontinuity in a function? Answer: A removable discontinuity is a point where the function is undefined, but it can be made continuous by defining the function at that point. A non-removable discontinuity is a point where the function cannot be made continuous.

9. What is the limit of a constant function? Answer: The limit of a constant function is equal to the constant value at any point.

10. Can a function have a limit that does not exist? Answer: Yes, a function can have a limit that does not exist if the function oscillates or jumps around the limiting value.

Calculus is a branch of mathematics that deals with the study of change, and it is essential for understanding a wide range of phenomena in science, engineering, and economics. One of the most important concepts in calculus is the notion of limits. In this article, we will discuss limits from a rigorous approach, including the definition of limits, the properties of limits, and how to evaluate limits using algebraic techniques.

### Definition of Limits:

The limit of a function f(x) as x approaches a is the value that f(x) approaches as x gets closer and closer to a. Mathematically, this is expressed as: lim f(x) = L x ? a where L is a real number or infinity, and the symbol ? means "approaches." This means that the value of f(x) may or may not be equal to L when x equals a, but as x gets arbitrarily close to a, f(x) gets arbitrarily close to L.

### Properties of Limits:

Limits have several important properties that make them useful in calculus. These properties include:
1. The limit of a sum is the sum of the limits:
lim (f(x) + g(x)) = lim f(x) + lim g(x) x ? a x ? a x ? a
1. The limit of a product is the product of the limits:
lim (f(x) * g(x)) = lim f(x) * lim g(x) x ? a x ? a x ? a
1. The limit of a quotient is the quotient of the limits (provided that the denominator does not equal zero):
lim (f(x) / g(x)) = lim f(x) / lim g(x) x ? a x ? a x ? a
1. The limit of a constant times a function is the constant times the limit of the function:
lim (c * f(x)) = c * lim f(x) x ? a x ? a

### To evaluate limits using algebraic techniques, we can use several methods, including:

1. Direct substitution: If the function is continuous at the point a, then we can substitute a for x and evaluate the function. If the function is not continuous, then we need to use other methods.
2. Factoring: We can factor the function and simplify the expression to eliminate any common factors and make the limit easier to evaluate.
3. Rationalizing: We can use algebraic techniques to remove any square roots or other radicals from the expression to simplify it and make it easier to evaluate.
4. Using L'Hopital's Rule: If we have an indeterminate form of the type 0/0 or infinity/infinity, we can use L'Hopital's rule to evaluate the limit. This involves taking the derivative of the numerator and denominator separately and then evaluating the limit again.
5. Using Squeeze Theorem: If we have a function that is bounded between two other functions, and the limits of the two functions are equal, then the limit of the bounded function is also equal to that limit.

### Conclusion:

In conclusion, limits are a fundamental concept in calculus, and they play a crucial role in many areas of mathematics and science. A rigorous approach to limits involves understanding the definition of limits, the properties of limits, and how to evaluate limits using algebraic techniques. By mastering these concepts, we can gain a deeper understanding of calculus and apply it to solve real-world problems.