Introduction to limits
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What does lim_(x → a) f(x) = L mean?
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5.1.18 cards
What does lim_(x → a) f(x) = L mean?
As x gets closer to a (from both sides), f(x) gets closer and closer to L. The limit does not depend on f(a).
Think: what does the graph HEAD TOWARDS near x = a?
How do you evaluate lim_(x → a) f(x) for a polynomial?
Direct substitution: replace x with a. E.g. lim_(x → 3)(2x+1) = 2(3)+1 = 7.
Polynomials have limits everywhere — just substitute.
What do you do when substitution gives 0/0?
Factor and cancel the common factor, then substitute. E.g. (x^2-4)/(x-2) = x+2, so the limit at x=2 is 4.
0/0 is a signal to factorise — never the final answer.
What is a one-sided limit?
lim_(x → a^-): approach from the LEFT (values below a). lim_(x → a^+): approach from the RIGHT (values above a).
The little - or + superscript shows direction.
When does the two-sided limit lim_(x → a) f(x) exist?
Only when both one-sided limits exist AND are equal: lim_(x → a^-) f(x) = lim_(x → a^+) f(x).
If left ≠ right, the limit does not exist (DNE).
Can lim_(x → a) f(x) = L even if f(a) is undefined?
YES. The limit only depends on values near a, not AT a. Example: (x^2-4)/(x-2) is undefined at x=2 but the limit is 4.
Limits and function values are different thing.
A table show: as x → 5, f(x) → 8 from both side. What is the limit?
lim_(x → 5) f(x) = 8. Read from the table: both sides converge to the same value.
Two sides must agree.
Evaluate lim_(x → 4) (x^2 - 16)/(x - 4).
Factor: x^2 - 16 = (x-4)(x+4). Cancel (x-4). Substitute x=4: 4+4 = 8. The limit is 8.
Spot the difference of two square.
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