2026 Fall Section 2.4 Limits Algebraic Method
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Overview
Araeipour reviews limit notation, one-sided limits, and asymptotes, then applies sum, difference, product, quotient, constant-multiple, power, and root laws to evaluate limits. The central method for indeterminate forms such as 0/0 is to simplify algebraically: factor rational expressions, clear complex fractions with a least common denominator, or rationalize radicals with a conjugate before substituting.
Key takeaways
- A two-sided function limit exists only when the left- and right-hand limits agree; the actual value f(a) may be different or undefined.
- Direct substitution is effective for continuous expressions, but 0/0 is indeterminate and requires simplification rather than an immediate answer.
- For rational 0/0 limits, factor numerator and denominator to expose and cancel the factor causing the denominator to vanish, then substitute.
- For complex fractions such as (1/4 + 1/x)/(x + 4), multiplying the inner numerator by its least common denominator 4x can expose the factor needed for cancellation.
- For differences involving square roots, multiply by the conjugate to use the difference-of-squares identity; this changes radical 0/0 forms into cancellable algebraic factors.
- Even-index roots require nonnegative inputs, while odd-index roots allow negative values.
Chapters
- A sequence converges when its limit is a finite number; limits of positive infinity, negative infinity, or no limit indicate divergence.
- For a function limit at x = a, the left- and right-hand values must approach the same number, regardless of whether f(a) equals that number or is defined.
- A finite limit as x approaches positive or negative infinity gives a horizontal asymptote y = L; an infinite one-sided or two-sided limit near x = a indicates a vertical asymptote x = a.
- When f(x) approaches L and g(x) approaches M, limits distribute across sums, differences, products, and quotients, with M required to be nonzero for the quotient law.
- Constant multiples, positive-integer powers, and roots can be handled by applying the corresponding operation to the function's limit; even roots require a nonnegative limit.
- For continuous functions, direct substitution gives the limit, as in x² - 3x + 7 approaching 11 when x approaches 4.
- Graph reading requires checking both sides: at x = -2 the displayed functions approach 1 and -1, while a mismatch in one-sided values at x = 1 makes the relevant product limit nonexistent.
- For x > 0, |x|/x = 1; for x < 0, |x|/x = -1, and the function is undefined at x = 0.
- As x approaches 0 from the left, |x|/x approaches -1, while from the right it approaches 1.
- Because the two one-sided limits differ, the two-sided limit of |x|/x at 0 does not exist.
- Direct substitution evaluates the example with x approaching 2 as 3/2; when substitution instead gives 0/0, further algebra is required.
- For (1/4 + 1/x)/(x + 4) as x approaches -4, multiplying the complex numerator by its least common denominator 4x reveals a cancellable x + 4 factor and yields -1/16.
- Factoring x² - 25 as (x - 5)(x + 5) and x² - 5x as x(x - 5) cancels the shared x - 5 factor; substitution then gives 2.
- A denominator that vanishes at x = a signals a factor associated with x - a, such as x + 4 when a = -4; cancel only after factoring.
- For (√(1 + h) - 1)/h as h approaches 0, multiply numerator and denominator by √(1 + h) + 1 to create a difference of squares and cancel h; the limit is 1/2.
- For (√(x² + 9) - 5)/(x + 4) as x approaches -4, multiplying by the conjugate turns the numerator into x² - 16.
- Factoring x² - 16 as (x - 4)(x + 4) cancels x + 4, and substituting -4 into the simplified expression gives -4/5.
Transcript excerpt
0:06 Good morning class. Here's chapter 2 limits section 2.4 limits algebraic methods. We've covered limits of sequences, limits of functions at infinity, limits of functions at finite numbers, and now limits algebraic methods. So here's the synopsis of what we have covered. We want to find the limit. First the definition. Limit of f of x as x approaches a= l. We read the limit of f of x as x approaches a= l. The limit of a subn as n approaches infinity is l. We say that a subn converges to l. If the limit of a subn as n approaches infinity is either
0:53 infinity, negative infinity or dn e does not exist, we say the sequence diverges. undefined limits basically plug in and observe. We have the following graphs and we want to discuss the similarities and differences. As you can see, we are interested in limit of f of x as x approaches a. So when we approach a from the left over here, we're moving this way moving up here from the right moving this way moving down. In both cases we are moving towards this point which has a y-coordinate l. And therefore the limit of f of x as x approaches a is l for all cases. What's the difference? In this case also f of a equals l. In this case f of a is something else b.
1:42 And in this case f of a is not given. Therefore it's undefined. Definition of a horizontal asmtote is as follows. If we take a limit of a function f ofx as x approaches either negative or positive infinity and we get l then the equation of a line y= l represents the horizontal asmtote of the graph. VA short for vertical asmtote. The limit
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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Ara Araeipour.