Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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3:00 minutes

Problem 2.34

Textbook Question

Evaluate each limit and justify your answer.
lim t→4 t−4 /√t−2

Verified step by step guidance

Identify the form of the limit as \( t \to 4 \). Notice that both the numerator \( t - 4 \) and the denominator \( \sqrt{t} - 2 \) approach 0, indicating an indeterminate form \( \frac{0}{0} \).

To resolve the indeterminate form, consider rationalizing the denominator. Multiply the numerator and the denominator by the conjugate of the denominator, \( \sqrt{t} + 2 \).

This gives: \( \frac{(t - 4)(\sqrt{t} + 2)}{(\sqrt{t} - 2)(\sqrt{t} + 2)} \). Simplify the denominator using the difference of squares formula: \( (\sqrt{t})^2 - 2^2 = t - 4 \).

The expression simplifies to \( \frac{(t - 4)(\sqrt{t} + 2)}{t - 4} \). Cancel the common factor \( t - 4 \) from the numerator and the denominator.

After canceling, evaluate the limit of the simplified expression \( \sqrt{t} + 2 \) as \( t \to 4 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. Evaluating limits often involves techniques such as substitution, factoring, or applying L'Hôpital's rule when dealing with indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they indicate the need for further analysis or manipulation of the expression to find the limit. Techniques such as algebraic simplification or L'Hôpital's rule are commonly used to resolve these forms.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the context of limits, rational functions often require careful analysis of their behavior as the variable approaches specific values, particularly when the denominator approaches zero. Understanding how to simplify these functions and identify removable discontinuities is essential for evaluating limits effectively.

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