1. Limit at a Point
Definition: The limit of a function f(x)f(x) as xx approaches a point aa is the value that f(x)f(x) approaches as xx gets closer to aa. It is denoted as:
limx→af(x)=L\lim_{x \to a} f(x) = L
If for every ϵ>0\epsilon > 0, there exists a δ>0\delta > 0 such that ∣f(x)−L∣<ϵ|f(x) - L| < \epsilon whenever 0<∣x−a∣<δ0 < |x - a| < \delta, then the limit exists and is equal to LL.
Right-hand limit: limx→a+f(x)\lim_{x \to a^+} f(x) is the value of f(x)f(x) as xx approaches aa from the right.
Left-hand limit: limx→a−f(x)\lim_{x \to a^-} f(x) is the value of f(x)f(x) as xx approaches aa from the left.
Existence of limit: For limx→af(x)\lim_{x \to a} f(x) to exist, the right-hand and left-hand limits must be equal.
2. Properties of Limits
Some key properties of limits are:
Sum Property:
limx→a(f(x)+g(x))=limx→af(x)+limx→ag(x)\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
Product Property:
limx→a(f(x)⋅g(x))=limx→af(x)⋅limx→ag(x)\lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)
Quotient Property:
limx→a(f(x)g(x))=limx→af(x)limx→ag(x),if limx→ag(x)≠0\lim_{x \to a} \left(\frac{f(x)}{g(x)}\right) = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \text{if} \ \lim_{x \to a} g(x) \neq 0
Scalar Multiple Property:
limx→ac⋅f(x)=c⋅limx→af(x)\lim_{x \to a} c \cdot f(x) = c \cdot \lim_{x \to a} f(x)
where cc is a constant.
Power and Root Property:
limx→af(x)n=(limx→af(x))n(if n is a positive integer)\lim_{x \to a} f(x)^n = \left(\lim_{x \to a} f(x)\right)^n \quad \text{(if \( n \) is a positive integer)}
3. Computation of Limits of Various Types of Functions
To compute limits, use the following methods:
Direct Substitution:
If f(x)f(x) is continuous at x=ax = a, then
limx→af(x)=f(a)\lim_{x \to a} f(x) = f(a)
Factorization:
If direct substitution results in an indeterminate form like 00\frac{0}{0}, factor the function and simplify.
Rationalization:
If the limit involves square roots, multiply by the conjugate to simplify the expression.
L'Hôpital's Rule:
If direct substitution gives 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, differentiate the numerator and denominator separately and then compute the limit.
4. Continuity at a Point
A function f(x)f(x) is continuous at a point x=ax = a if:
f(a)f(a) is defined.
limx→af(x)\lim_{x \to a} f(x) exists.
limx→af(x)=f(a)\lim_{x \to a} f(x) = f(a).
If any of these conditions fail, the function is discontinuous at x=ax = a.
5. Continuity Over an Interval
A function is continuous over an interval II if it is continuous at every point in II. Specifically:
Continuous on (a,b)(a, b): The function is continuous for all x∈(a,b)x \in (a, b).
Continuous on [a,b][a, b]: The function is continuous on (a,b)(a, b), and it is also continuous at the endpoints aa and bb.
6. Intermediate Value Theorem
The Intermediate Value Theorem states that if f(x)f(x) is continuous on the closed interval [a,b][a, b], and NN is any number between f(a)f(a) and f(b)f(b), then there exists some c∈[a,b]c \in [a, b] such that:
f(c)=Nf(c) = N
In other words, a continuous function takes every value between its minimum and maximum on a closed interval.
7. Types of Discontinuities
Discontinuities can be classified into several types:
Jump Discontinuity:
The left-hand and right-hand limits exist but are not equal. For example, a piecewise function with different values on either side of a point.
Infinite Discontinuity:
One or both of the limits approach infinity. This happens when the function has vertical asymptotes.
Removable Discontinuity:
The left-hand and right-hand limits exist and are equal, but the function is not defined at the point, or the function value at the point does not match the limit.
This overview covers key concepts related to Limits and Continuity. Let me know if you'd like further explanation on any topic!
Here are some key questions related to Limits and Continuity:
Question: Evaluate limx→2(3x+5)\lim_{x \to 2} (3x + 5).
Question: If limx→1f(x)=4\lim_{x \to 1} f(x) = 4 and limx→1g(x)=3\lim_{x \to 1} g(x) = 3, find the following:
(a) limx→1(f(x)+g(x))\lim_{x \to 1} (f(x) + g(x))
(b) limx→1(f(x)⋅g(x))\lim_{x \to 1} (f(x) \cdot g(x))
(c) limx→1f(x)g(x)\lim_{x \to 1} \frac{f(x)}{g(x)} (if g(x)≠0g(x) \neq 0)
Question: Determine if the function f(x)=x2−4x−2f(x) = \frac{x^2 - 4}{x - 2} is continuous at x=2x = 2.
Question: Is the function f(x)=1xf(x) = \frac{1}{x} continuous on the interval [1,3][1, 3]?
Question: Given f(x)=x3−4xf(x) = x^3 - 4x, find the value of cc such that f(c)=0f(c) = 0 in the interval [1,2][1, 2], if possible.
Question: Classify the discontinuity of the function f(x)=1xf(x) = \frac{1}{x} at x=0x = 0.
Question: Compute limx→0sin(x)x\lim_{x \to 0} \frac{\sin(x)}{x} using L'Hôpital's Rule.
Let me know if you'd like solutions to any of these questions!
Here’s a concise version of Limits & Continuity:
Definition: limx→af(x)=Lif for every ϵ>0, ∃δ>0 such that ∣f(x)−L∣<ϵ when ∣x−a∣<δ\lim_{x \to a} f(x) = L \quad \text{if for every} \ \epsilon > 0, \ \exists \delta > 0 \ \text{such that} \ |f(x) - L| < \epsilon \ \text{when} \ |x - a| < \delta
Right-hand limit: limx→a+f(x)\lim_{x \to a^+} f(x)
Left-hand limit: limx→a−f(x)\lim_{x \to a^-} f(x)
Sum: lim(f(x)+g(x))=limf(x)+limg(x)\lim (f(x) + g(x)) = \lim f(x) + \lim g(x)
Product: lim(f(x)⋅g(x))=limf(x)⋅limg(x)\lim (f(x) \cdot g(x)) = \lim f(x) \cdot \lim g(x)
Quotient: limf(x)g(x)=limf(x)limg(x)\lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)} if limg(x)≠0\lim g(x) \neq 0
Direct Substitution: If f(x)f(x) is continuous, limx→af(x)=f(a)\lim_{x \to a} f(x) = f(a).
Factorization: Simplify by factoring and then substituting.
Rationalization: Use the conjugate for limits involving square roots.
L'Hôpital's Rule: Use when you get 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}.
A function f(x)f(x) is continuous at x=ax = a if:
f(a)f(a) is defined.
limx→af(x)\lim_{x \to a} f(x) exists.
limx→af(x)=f(a)\lim_{x \to a} f(x) = f(a).
A function is continuous over an interval if it's continuous at every point in the interval.
If f(x)f(x) is continuous on [a,b][a, b], and NN is between f(a)f(a) and f(b)f(b), then there exists c∈[a,b]c \in [a, b] such that f(c)=Nf(c) = N.
Jump Discontinuity: Limits from left and right exist but are unequal.
Infinite Discontinuity: Limits approach infinity.
Removable Discontinuity: Limits exist and are equal, but the function is not defined or doesn't match the limit at the point.
This is a compact overview of the main points. Let me know if you need further details!