Calculus Midterm 1 Formulas

Basic Properties of Limits & Limits towards infinity

Limits towards infinity: 

• if limit is a polynomial then you can take the leading term 

• when limits are rational functions you can also take the leading term 

Addition

Lim x→a (f(x) + g(x))=

lim x→a f(x) + lim x→a (g)x

Lim x→∞(f(x) + g(x))=

lim x→∞ f(x) + lim x→∞ (g)x

Subtraction

lim x→a (f(x)-g(x)) =

lim x→a f(x) - lim x→a g(x)

lim x→∞ (f(x)-g(x)) =

lim x→∞ f(x) - lim x→∞ g(x)

Multiplication

lim x→a (f(x)g(x)) =

(lim x→a f(x))(lim x→a g(x))

lim x→∞ (f(x)g(x)) =

(lim x→∞ f(x))(lim x→∞ g(x))

Division

lim x→a f(x)/g(x) =

lim x→a f(x) / lim x→a g(x) 

BUT g(x) can not = 0

lim x→∞ f(x)/g(x) =

lim x→∞ f(x) / lim x→∞ g(x) 

BUT g(x) can not = 0

Exponent

lim x→a f(x)ⁿ = (lim x→a f(x))ⁿ

lim x→∞ f(x)ⁿ = (lim x→∞ f(x))ⁿ

Square root

lim x—>a n√f(x) = n√lim x→a f(x)

when f(x) is > 0

lim x—>∞ n√f(x) = n√lim x→∞ f(x)

when f(x) is > 0

f(x) and a constant (for infinity only)

lim x→∞ kf(x) = k (lim x→ ∞ f(x))

Constant only (for only infinity)

lim x→ ∞ k=k

Operations with infinity

Addition

+∞ + ∞ = +∞

Subtraction

-∞ - ∞ = -∞

Multiplication (positives)

(+∞)(+∞) = +∞ 

Multiplication (negatives)

(-∞)(-∞) = +∞

Multiplication (positive & negative)

(+∞)(-∞) = -∞

Infinity Rules & Indeterminate forms

1/0^- = -∞

1/0⁺ = +∞

± ∞ / ± ∞ = IND

0^± / 0^± = IND

0(± ∞) = IND

0⁰ = IND

∞⁰ = IND

1∞ = IND

+∞ - ∞ = IND

Limits of exponentail functions

lim x→ -∞ e^x = 0

lim x→ +∞ e^x = +∞

lim x→ +∞ 1/e^x = 0

lim x→ -∞ 1/e^x = +∞

lim x→ -∞ e^-x = +∞

lim x→ +∞ e^-x = 0

lim x→ 0+ lnx = -∞

lim x→ +∞ lnx = +∞

lim x→ -∞ lnx = UNDEFINED b/c does not exist

ln (0) = DNE

ln (1) ln(-1) ln l-1l = 0

Properties of continous functions

• can draw a graph without lifting your pencil- no jumps or holes

• Polynomials are continous at every point

• Rational functions are continous at every point as long as the denominater does not equal 0

• f(x) is continous at point x=c

  • both left and right limit exist and c is defined 

• Intermediate value property 

  • If you start at f(a) and end at f(b) you must through every possible value between that interval

If f(x) and g(x) are both continous at point c then..

Addition

f(x) = g(x) is continous at C

Subtraction

f(x) - g(x) is continous at C

Multiplication

f(x) x g(x) is continuous at C 

Division

f(x) / g(x) is continuous at C as long as g(c) does not equal 0

Discontinuity 

Jump Discontinuity

• left and right limit both exist but are not equal 

Removable Discontinuity

• limits exists but either it is not defined at f(a) or f (a) doesn’t equal f(x) 

Infinite Discontinuity 

• as x→a the function is heading towards infinity 

Trigonometric Functions

f₁(x) = sinx

f₂(x) = cosx

Domain is all Real and range is (-1,1)

f₃(x) = tanx

f₄(x) = secx = 1/cosx

Domain is all Real - x=0 ex. (π/2, 3π/2)

Range is all Real

f₅(x) = cscx

f₆(x) = cotx

Domain is all real - x=0 (ex. π, 2π)

ln:

Domain- (0, ∞)

Range- (-∞, ∞)

Inverse of these functions the range becomes the domain

Squeeze Theorem

g(x) ≤ f(x) ≤ h(x)

If lim x→a g(x) = lim x→a h(x) = L

then lim x→a f(x) = L

IMPORTANT TO REMEMBER:

-x ≤ sin(1/x) ≤ x

Key Limits of Squeeze Theorem

lim x→+∞ 1/x= 0

lim x→ -∞ 1/x= 0

lim x→0+ 1/x= +∞

lim x→0- 1/x = -∞

lim x→0 sinx/x = 1

lim x→0 x/sinx = 1

lim x→0 sin(ax)/ax = 1

lim x→0 1-cosx/x = 0 

Important functions for derivatives

Equation of a line:

y - f(x) = m(x-a)

m= f(x+h) - f(x) / a

Velocity:

V_avg = change in position / time elapsed = f(t₀ + h) - f(t₀) / h

V_int = lim h→0 f(t₀ +h) - f(t₀) / h

if velocity is negative it is going backwards

Rate of Change:

r_avg = f(x₁) - f(x₀) / x₁ - x₀

r_int = lim x₁ → x₀ f(x₁) - f(x₀) / x₁ - x₀

lim x₁ → x ₀ represents that is it close to some value

The Derivative

f¹(x) = lim h→0(±) f(x+h) - f (x) / h

if the limit exists at some point x then it is differentiable at point x

*The derivative is the slope of the tangent line*

Differentiability & Continuity 

differentiable means that the derivative of the function exists and the function has a well-defined tangent line at that point

differentiable —> continuous 

Continuous does not always mean differentiable 

Steps to prove Continouity & Differentiability

Continuity:

1) is f(a) defined 

2) does f(a) exists from the left and right of the limit 

3) does the overal limit of f(a) exist (if yes then it is continuous b/c left & right match up) 

Differentiability:

1) Check if limit is continuous at that point 

2) Check if the limit exists overall h→0 

3) Check if the limit exists from the left and the right h→0± (if it does and it differentiable at the point that it is continuous)