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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2:13 minutes

Problem 27

Textbook Question

Determine the following limits.
lim x→5 x − 7 / x(x − 5)^2

Verified step by step guidance

Identify the type of limit problem: This is a limit as \( x \) approaches 5, and the expression is \( \frac{x - 7}{x(x - 5)^2} \).

Check for direct substitution: Substitute \( x = 5 \) into the expression to see if it results in an indeterminate form. \( \frac{5 - 7}{5(5 - 5)^2} = \frac{-2}{0} \), which is undefined.

Recognize the indeterminate form: Since the denominator becomes zero, this is a case of an indeterminate form, suggesting the need for further analysis.

Consider simplifying or factoring: Since direct substitution leads to an undefined form, consider simplifying the expression or using algebraic manipulation to resolve the indeterminate form.

Apply L'Hôpital's Rule if applicable: If the limit results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule can be used by differentiating the numerator and the denominator separately and then taking the limit again.

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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. In this case, we are interested in the limit as x approaches 5, which requires evaluating the function's behavior close to that point.

Indeterminate Forms

Indeterminate forms occur in limit problems when direct substitution leads to expressions like 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation, L'Hôpital's Rule, or other techniques to resolve the limit. In the given limit, substituting x = 5 results in an indeterminate form, necessitating additional steps to find the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, specifically 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule can simplify the process of finding limits, especially when direct evaluation is complex.

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