Limits: The Building Block of Calculus

1. Steps to finding a limit2. Limits approaching a value3. One-Sided limits4. Limits at infinity5. Limits with radicals 6. Trig Limits7. Limits of ln(x) and ex

1- Steps:Direct Substitution

Outcomes:A number means - continuousNumerator and denominator of a rational

function both = 0 - not continuous(hole)Numerator is a real #, but the denominator is 0.

(approaches an vertical asymptote)

2- Limits Approaching a Value

Plug in the numberExample: x2-9

→ (2)2-9=-5 → This equation is a continuous function

2limx

The End…Katherine Pistorius

Meghan Weisel

2- Limits Approaching a ValuePlug in the numberExample 2: (x2-4)/(x-2)

→ (4-4)/(2-2) =0/0

→ Factor Top and Cancel → Plug in 2 → Hole (2,4)

2limx

2- Limits Approaching a ValuePlug in the numberExample 3:

x/(x-1)

→ 1/(1-1) = 1/0

→ Vertical asymptote at x=1→ DNE

1lim

x

3- One-Sided Limits Used to describe the value of the function as

x approaches a specified value from a given direction.

Example 1:

x/(x-1)

→Graph → + Infinity

lim1x

4- Limits at InfinityDegree n < Degree d y=0Degree n > Degree d No asymptoteDegree n = Degree d Leading Coefficient (n/d)

Example: (2x2-3x+1)/(5x2+2x-3)

→leading coefficients are equal→ The answer is 2/5

limx

5- limits with RadicalsThe limit of an nth root is the nth root of the limit

Example:

→ put the limit inside of the 3rd root → answer:

limx

3 )86/()53( xx

6- Trig Limits

Example: cos(1/x)

cos (1/x) cos(0)

→ the answer is: 1

xlim

→

→

7- Limits of lnx and ex

ln(x)= +∞ ln(x) = -∞ ln(x)=DNE

ex = +∞ ex = 0

xlim

0limx 0

limx

xlim

xlim