# 1.2 Limits Analyticallyap Calculus

##### Theorem1.3.1Basic Limit Properties

Let (btext{,}) (ctext{,}) (L) and (K) be real numbers, let (n) be a positive integer, and let (f) and (g) be functions with the following limits: begin{align*}lim_{xto c}f(x)amp=Lamplim_{xto c} g(x)amp = Ktext{.}end{align*}

Given a function y = f. ( x) and an x -value, c, we say that “the limit of the function f, as x approaches c, is a value L ”: 1. If “ y tends to L ” as “ x tends to c.”. If “ y approaches L ” as “ x approaches c.”. If “ y is near L ” whenever “ x is near c.”. Limits describe how a function behaves near a point, instead of at that point. This simple yet powerful idea is the basis of all of calculus. If you're seeing this message, it means we're having trouble loading external resources on our website. 1.2 Limits Analytically Name: Write your questions and thoughts here! Calculus Recall: What is a limit? Notes Finding a limit: 1. Direct Substitution Factor and Cancel 1. 5: T 62 T F4; 2. Lim → 6 √3 F2 3. Lim → 4 4 65 T T 4. Lim →?; 2 6 E13 F7 T E7 Rationalize Two variables 5.

### 1.2 Limits Analyticallyap Calculus Frq

The following limits hold.

Constant

(limlimits_{xto c} b = b)

Identity

(limlimits_{xto c} x = c)

Sum/Difference

(limlimits_{xto c}(f(x)pm g(x)) = Lpm K)

Scalar Multiple

(limlimits_{xto c}(bcdot f(x)) = bL)

Product

(limlimits_{xto c} (f(x)cdot g(x)) = LK)

Quotient

(limlimits_{xto c} (f(x)/g(x)) = L/Ktext{,}) when (Kneq 0)

Power

(limlimits_{xto c} f(x)^n = L^n)

Root

(limlimits_{xto c} sqrt[n]{f(x)} = sqrt[n]{L})

(If (n) is even, (L) must be non-negative.)

Composition

If either of the following holds:

1. (limlimits_{xto c}f(x)=Ltext{,}) (limlimits_{xto L}g(x)=Ktext{,}) and (g(L)=K)

2. (limlimits_{xto c}f(x)=Ltext{,}) (limlimits_{xto L}g(x)=Ktext{,}) and (f(x)neq L) for all (x) close to but not equal to (c)

then (limlimits_{xto c}g(f(x)) = Ktext{.})