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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?
- It does not exist.
- It equals the constant itself. (correct)
- It equals zero.
- It approaches infinity.
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)?
- Limit of a sum
- Limit of a quotient
- Limit of a difference
- Constant multiple rule (correct)
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?
- lim- f(x) must equal lim+ f(x). (correct)
- lim+ f(x) must be greater than zero.
- lim- f(x) must be less than lim+ f(x).
- lim- f(x) must equal zero.
What strategy is typically used to resolve indeterminate forms like 0/0?
What strategy is typically used to resolve indeterminate forms like 0/0?
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?
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?
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?
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?
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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
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