### Limits - Limits describe the behavior of a function as its input approaches a particular value. - The notation lim_x→a f(x) = L means that as x gets closer and closer to a (from both sides), f(x) gets closer and closer to L. - Limits do not concern the function's value at the specific point 'a', but rather its behavior around that point. - The function does not need to be defined at 'a' for the limit to exist. - To estimate a limit, evaluate the function at values of x approaching 'a'. ### One-Sided Limits - Right-handed limit: lim_x→a⁺ f(x) = L, describes the behavior of f(x) as x approaches 'a' from values greater than 'a'. - Left-handed limit: lim_x→a⁻ f(x) = L, describes the behavior of f(x) as x approaches 'a' from values less than 'a'. - A limit exists if both one-sided limits at 'a' are equal to the same value. ### Limits at Infinity - Limit at infinity describes the behavior of a function as its input becomes arbitrarily large (positive infinity) or arbitrarily small (negative infinity). - lim_x→∞ f(x) and lim_x→⁻∞ f(x) indicate behavior of f(x) at large positive or negative values for x. - If f(x) has a horizontal asymptote at y = L, then either lim_x→∞ f(x) = L or lim_x→⁻∞ f(x) = L (or both). ### Key Properties of Limits - Limits are often affected in predictable ways by arithmetic operations, e.g., addition, subtraction, multiplication, division - The limit of a constant times a function is the constant times the limit of the function. - The limit of a sum is the sum of the limits. - The limit of a difference is the difference of the limits. - The limit of a product is the product of the limits. - The limit of a quotient is the quotient of the limits, provided the denominator is non-zero. - When the limiting value is a limit of x tending to infinity and x to a power r is in the denominator and r is positive the limit is 0. ### Polynomials at limits at infinity - For a polynomial of degree 'n', the limit as x approaches either positive or negative infinity is determined entirely by the term with the highest power of x. The limit is equivalent to the limit of that largest power term. ### Examples - Examples involving various functions (polynomials, trigonometric functions, piecewise functions) illustrate these concepts, demonstrating how limits can and cannot be computed or approximated.

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Questions and Answers

What is the primary focus when evaluating limits at a point?

Which of the following statements is true regarding the calculation of limits?

What does the notation lim $x \to a$ represent in the context of limits?

In the limit notation lim $t \to 5$ of the function $t^3 - 6t^2 + 25t - 5$, what is being evaluated?

Why are some methods not typically used for computing limits?

What might be the result of taking limits improperly?

What general aspect of limits is emphasized in the discussed section?

When approaching limits graphically, what aspect is considered?

What does the limit at a point depend on?

What conclusion can be drawn about the function value and the limit at a point?

Which method is discouraged for estimating limits?

Why can tables of values sometimes be misleading for limits?

What characteristic must a function exhibit for a limit to exist?

Under what circumstance can the limit of a function not exist?

What issue arises from using graphs to determine limits?

Why is it important to understand the concept of limits?

In which of the following scenarios will a limit be undetermined?

What kind of limit is represented by the function H(t) in the example?

What must be true for the function cos(πt) at the limit as t approaches 0?

What is a potential drawback of estimating limits graphically?

Which example indicates that limits can exist even when the function is undefined?

What happens to the limit of the function H(t) as t approaches 0 from the right?

Which statement accurately describes right-handed and left-handed limits?

When can we say that the normal limit exists for a function at a point?

What will be the result if the right-handed limit and the left-handed limit at a point are not equal?

What is required for a one-sided limit to exist?

In the case of H(t), what is the left-handed limit as t approaches 0?

Which of the following is true regarding the one-sided limits of a function?

What is the notation used for the right-handed limit at a?

In what scenario can a normal limit be determined to exist?

What does it imply if lim x → a+ f(x) ≠ lim x → a− f(x)?

Which of the following is an example of a function that might have different one-sided limits at a point?

How do you denote the left-handed limit approaching a?

If both one-sided limits of a function exist but are not equal, what can be concluded?

In the context of limits, what does the term 'settle down' mean?

What can be concluded if the limit of a function exists as x approaches a?

Which of the following statements about limits is true?

In estimating limits, what is the purpose of plugging in values of x approaching a?

If f(x) is larger than L, which inequality represents making f(x) close to L by a margin of 0.001?

What is a key detail to remember when evaluating limits?

What does the notation lim x→a f(x) = L signify?

Why are limits important in calculus?

When analyzing the limit of a function as x approaches a, what should be considered?

What does the presence of an open dot at a point on a graph signify in relation to limits?

What must be true for the limit to be considered valid?

If the limit does not exist as x approaches a, which of the following could be true?

In terms of limit notation, what does f(x) → L as x → a imply?

Which statement accurately describes the relationship between limits and continuity?

What is an example of a situation where a limit can exist even if the function is not defined at that point?

What happens to the limit of a function as x approaches infinity if it is in the form $c x^r$, where r is a positive rational number?

How does the sign of the constant c in the limit $lim_{x→∞} c x^r$ affect the behavior of the limit?

When evaluating limits of polynomials at infinity, which term dominates the behavior of the polynomial?

For the polynomial $p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, what is true about its limit as x approaches negative infinity?

Which of the following statements is true regarding horizontal asymptotes?

What assumption should not be made about limits at infinity based on previous examples?

In the context of limits, what defines the behavior of the function $f(x)$ near vertical asymptotes?

What does the limit $lim_{x→∞} \sqrt{3x^2 + 6}/(5 - 2x)$ indicate about the function's behavior?

When considering the limit $lim_{x→−∞} \sqrt{3x^2 + 6}/(5 - 2x)$, what will be the result?

What do the terms of a polynomial dictate about the limit as x approaches negative infinity?

In polynomial functions, which will occur as x becomes very large and positive for the limit $lim_{x→∞} p(x)$?

Which of the following is NOT a property that affects the limit as x approaches infinity for rational functions?

What conclusion can be drawn about two limits at infinity having the same finite value?

Questions and Answers

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Limit

Limit notation

Limit estimation

Function values

Limit at a point

Limit and function value difference

Estimating limits (tables)

Limit existence

Limit non-existence