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The main topic of this content is the fundamental concepts of limits in calculus, including their application in evaluating the behavior of functions as they approach specific values or infinity. It covers direct substitution for rational functions, the Quotient Rule, vertical asymptotes, one-sided limits, and algebraic techniques for indeterminate forms. Additionally, it discusses the limits of piecewise and exponential functions, emphasizing the importance of continuity and one-sided limits in determining the overall limit of a function.

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## Definition of Limits

### Importance of Limits

Limits are crucial for understanding the behavior of functions as they approach a particular value

### Rational Functions

Direct Substitution

Limits of rational functions can often be found by directly substituting the value into the function

More Complex Techniques

When evaluating limits at points outside the domain or at infinity, more complex techniques may be required

### Quotient Rule for Limits

The Quotient Rule is applicable for evaluating limits at infinity, as long as the limits of the numerator and denominator are finite

## Techniques for Evaluating Limits

### Algebraic Manipulation

Algebraic techniques such as polynomial division, factoring, and rationalizing can be used to simplify expressions and resolve indeterminate forms

### Vertical Asymptotes

Vertical asymptotes indicate discontinuities and require consideration of one-sided limits

### Piecewise Functions

Piecewise functions require careful evaluation of one-sided limits at points of definition change

## Continuity and Exponential Functions

### Continuity of Exponential Functions

Exponential functions, such as e^x, are continuous everywhere, simplifying limit evaluation

### Composite Exponential Functions

For composite exponential functions, the limit of the inner function must be determined before applying the exponential

## Strategies for Calculating Limits

### Limit Laws

A thorough understanding of Limit Laws is necessary for accurately calculating limits

### Algebraic Manipulation

Algebraic manipulation is often necessary to simplify expressions and address indeterminate forms

### One-Sided Limits

One-sided limits are crucial for evaluating piecewise functions and functions with discontinuities

Algorino

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