Limits form the cornerstone of calculus, embodying the value a function approaches as its input nears a specific point. This concept is pivotal in defining function behavior at particular points, especially where the function might not be explicitly defined.

The Concept of a Limit

• Defines the behavior of $f(x)$ as $x$ approaches a point $c$, aiming for a real number $R$.
• Crucial for analyzing functions at undefined points or exhibiting unusual behavior.

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Worked Examples

Example 1: Direct Substitution

Find $\lim_{x \to 4} (3x + 2)$

Substitute $x$ with 4:

Example 2: Factoring and Simplifying

Find $\lim_{x \to 1} \dfrac{x^2 - 1}{x - 1}$

1. Factor the numerator:

2. Cancel common terms:

3. Substitute $x$ with 1:

Example 3: Graphical Interpretation

Estimate $\lim_{x \to 0^+} \dfrac{1}{x}$.

1. As $x$ approaches 0 from the right, $\frac{1}{x}$ grows infinitely large.

2. The function does not approach a finite number but instead goes towards infinity:

Importance of Limits

• Foundation for Calculus: Enables understanding of derivatives, integrals.
• Defines Continuity: A function is continuous at a point if the function's limit at that point equals its value.
• Analyzes Asymptotic Behavior: Crucial for examining function behavior near undefined points or as functions approach infinity.

Practice Questions

Question 1

Find $\lim_{x \to 3} (2x^2 - 5x + 1)$.

Question 2

Calculate $\lim_{x \to -2} \dfrac{x^2 + 3x + 2}{x + 2}$.

Question 3

Determine $\lim_{x \to 0} \dfrac{\sin(x)}{x}$.

Solutions to Practice Questions

Solution to Question 1

Find $\lim_{x \to 3} (2x^2 - 5x + 1)$.

• Step 1: Substitute $x = 3$ into the function.

• Step 2: Perform the calculation.

• Conclusion:

Solution to Question 2

Calculate $\lim_{x \to -2} \dfrac{x^2 + 3x + 2}{x + 2}$.

• Step 1: Factor the numerator and simplify the expression.

• Step 2: Cancel the common factor $(x + 2)$.

• Step 3: Substitute $x = -2$ into the simplified function.

• Conclusion:

Solution to Question 3

Determine $\lim_{x \to 0} \dfrac{\sin(x)}{x}$.

• Recognize this as a fundamental limit in calculus.

• Conclusion: