Podcast
Questions and Answers
What is the limit of a constant function as x approaches any value?
What is the limit of a constant function as x approaches any value?
Which property of limits is illustrated by the equation lim (kf(x)) = k lim f(x)?
Which property of limits is illustrated by the equation lim (kf(x)) = k lim f(x)?
When determining if a limit exists using one-sided limits, what condition must be satisfied?
When determining if a limit exists using one-sided limits, what condition must be satisfied?
What strategy is typically used to resolve indeterminate forms like 0/0?
What strategy is typically used to resolve indeterminate forms like 0/0?
Signup and view all the answers
In the context of limits at infinity, what happens to the limit of a function if it approaches a negative infinity?
In the context of limits at infinity, what happens to the limit of a function if it approaches a negative infinity?
Signup and view all the answers
How should one choose the function to evaluate when calculating a left-hand limit for a piecewise function?
How should one choose the function to evaluate when calculating a left-hand limit for a piecewise function?
Signup and view all the answers
What is the result when evaluating lim (f(x)/g(x)) if lim g(x) is not equal to zero?
What is the result when evaluating lim (f(x)/g(x)) if lim g(x) is not equal to zero?
Signup and view all the answers
For the property lim (f(x) ± g(x)) = lim f(x) ± lim g(x), which of the following is true?
For the property lim (f(x) ± g(x)) = lim f(x) ± lim g(x), which of the following is true?
Signup and view all the answers
Study Notes
Limits
- When f(x) = k where k is a constant, then lim 𝑓(𝑥) = lim 𝑘 = 𝑘 𝑥→𝑎 𝑥→𝑎
- When 𝑓(𝑥) = 𝑥 𝑛 , then lim 𝑓(𝑥) = lim 𝑥 𝑛 = 𝑎𝑛 𝑥→𝑎 𝑥→𝑎
- lim 𝑘𝑓(𝑥) = 𝑘 lim 𝑓(𝑥) 𝑥→𝑎 𝑥→𝑎
- lim 𝑓 𝑥 ± 𝑔 𝑥 = lim 𝑓(𝑥) ± lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
- lim 𝑓 𝑥 𝑔 𝑥 = lim 𝑓(𝑥) lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
- lim 𝑓 𝑥 lim 𝑓 𝑥 𝑥→𝑎 lim = , lim 𝑔(𝑥) ≠ 0 𝑥→𝑎 𝑔 𝑥 lim 𝑔(𝑥) 𝑥→𝑎 𝑥→𝑎
Indeterminates and Limits
- 0/0 is an indeterminate form, it can be solved using factorization, or multiplying with conjugates, especially when dealing with surds.
One-Sided Limits
- When x approaches a from the left, it is denoted as x → a− (e.g. 0− = ‒0.000000…01)
- When x approaches a from the right, it is denoted as x → a+ (e.g. 0+ = 0.000000…01)
Determining Existence of a Limit
- For a limit to exist, the left-hand limit must equal the right-hand limit. lim− 𝑓(𝑥) = lim+ 𝑓(𝑥) = 𝐿 𝑥→𝑎 𝑥→𝑎
- This would then imply that lim 𝑓(𝑥) = 𝐿 𝑥→𝑎
- For piecewise functions, when finding the limit as x approaches a from the left, use the function defined for values less than a. When finding the limit as x approaches a from the right, use the function defined for values greater than a.
Infinite Limits
- When y approaches positive infinity from the right(0+), it is denoted as +∞.
- When y approaches negative infinity from the right(0+), it is denoted as -∞.
- When y approaches positive infinity from the left(0-), it is denoted as -∞.
- When y approaches negative infinity from the left(0-), it is denoted as +∞.
Limits at Infinity
- When y approaches positive or negative infinity, then the limit is equal to 0.
- lim 𝑦 = 0 ±∞
Infinite Indeterminates and Limits
- ∞ / ∞ is an indeterminate form and can be solved by dividing every term in the numerator and denominator by the highest power of x in the denominator.
lim = lim
𝑥→−∞ 𝑥2 + 1 𝑥→−∞ 𝑥2 1
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
This quiz covers key concepts related to limits in calculus, including the evaluation of limits for constant and polynomial functions, one-sided limits, and indeterminate forms. Test your understanding of these fundamental principles and see how well you can apply them in different scenarios.