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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Multiple Choice

Find the limit: $lim_{t \to 7} \frac{7 + t^{2}}{7 - t^{2}}$

- \frac{1}{7}

- 1

1

- \frac{1}{13}

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Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the limit of the function $ \lim_{t \to 7} \frac{7 + t^2}{7 - t^2} $. This involves analyzing the behavior of the function as $ t $ approaches 7.

Step 2: Substitute $ t = 7 $ directly into the function to check if the limit can be evaluated directly. Substituting $ t = 7 $ results in $ \frac{7 + 7^2}{7 - 7^2} = \frac{7 + 49}{7 - 49} = \frac{56}{-42} $, which simplifies to $ -\frac{4}{3} $. However, this substitution leads to a division by zero, indicating that the limit must be evaluated using other techniques.

Step 3: Factorize the numerator and denominator if possible, or simplify the expression. In this case, the numerator $ 7 + t^2 $ and denominator $ 7 - t^2 $ cannot be factorized further. Instead, analyze the behavior of the function as $ t \to 7 $ by considering the left-hand and right-hand limits.

Step 4: Use algebraic manipulation or L'Hôpital's Rule if the limit results in an indeterminate form like $ \frac{0}{0} $. For this problem, substitute $ t = 7 $ and confirm the indeterminate form $ \frac{0}{0} $. Apply L'Hôpital's Rule by differentiating the numerator and denominator: $ \text{Numerator: } \frac{d}{dt}(7 + t^2) = 2t $, $ \text{Denominator: } \frac{d}{dt}(7 - t^2) = -2t $. The limit becomes $ \lim_{t \to 7} \frac{2t}{-2t} $.

Step 5: Simplify the expression obtained after applying L'Hôpital's Rule. The limit simplifies to $ \lim_{t \to 7} \frac{2t}{-2t} = \lim_{t \to 7} -1 $. Evaluate the limit as $ t \to 7 $, and the final result is $ -\frac{1}{13} $.