1.1 – Limits: A Numerical and Graphical Approach Function Review 𝑓 𝑥 =2𝑥−7 𝑓 𝑥 =2𝑥−7 𝑓 3 = ? 𝑓 −5 = ? 𝑓 3 =6−7 𝑓 −5 =2 −5 −7 𝑓 3 =2 3 −7 𝑓 −5 =−10−7 𝑓 3 =−1 𝑓 −5 =−17 3,−1 −5, −17

1.1 – Limits: A Numerical and Graphical Approach Function Review 𝑓 𝑥 = 𝑥 2 +2𝑥−7 𝑓 −7 = ? 𝑓 −7 = −7 2 +2 −7 −7 𝑓 −7 =49−14−7 𝑓 −7 =28 −7, 28

1.1 – Limits: A Numerical and Graphical Approach Defn: Limit As the variable x approaches a certain value, the variable y approaches a certain value. lim 𝑥→𝑐 𝑓 𝑥 =𝐿 Find the requested limits from the graph of the given function. lim 𝑥→2 𝑓 𝑥 = 3

1.1 – Limits: A Numerical and Graphical Approach Defn: Limit As the variable x approaches a certain value, the variable y approaches a certain value. lim 𝑥→𝑐 𝑓 𝑥 =𝐿 Find the requested limit of the given function. 𝑓 𝑥 = 𝑥 2 −1 𝒙 𝒇 𝒙 1.9 2.61 lim 𝑥→2 𝑓 𝑥 1.99 2.9601 1.999 2.9960 lim 𝑥→2 𝑓 𝑥 = 3 2 ? 2.001 3.0040 2.01 3.0401 2.1 3.41

1.1 – Limits: A Numerical and Graphical Approach Find the requested limits for the given function. 𝑓 𝑥 = 𝑥 2 −1 lim 𝑥→−1 𝑓 𝑥 = −1 2 −1= lim 𝑥→0 𝑓 𝑥 = 0 2 −1= −1 lim 𝑥→2 𝑓 𝑥 = 2 2 −1= 3

1.1 – Limits: A Numerical and Graphical Approach Given the following graph of a function, find the requested limit. lim 𝑥→2 𝑔 𝑥 = 4 𝑔 2 = 6 2,6

1.1 – Limits: A Numerical and Graphical Approach Given the following graph of a function, find the requested limits. lim 𝑥→0 𝑓 𝑥 = 2 𝑓 0 = 2 0,2 lim 𝑥→−1 𝑓 𝑥 = 𝐷𝑁𝐸 𝑓 −1 = 2 −1,2 lim 𝑥→1 𝑓 𝑥 = 2 𝑓 1 = 3 1,3 lim 𝑥→2 𝑓 𝑥 = 𝐷𝑁𝐸 𝑓 2 = 3 2,3

1.1 – Limits: A Numerical and Graphical Approach A limit of a function can be analyzed from the left and right sides of a particular value of x. lim 𝑥→ 2 − 𝑔 𝑥 = 4 lim 𝑥→ 2 + 𝑔 𝑥 = 4 therefore lim 𝑥→2 𝑔 𝑥 = 4

1.1 – Limits: A Numerical and Graphical Approach A limit of a function can be analyzed from the left and right sides of a particular value of x. lim 𝑥→ 1 − 𝑓 𝑥 = 2 lim 𝑥→ 1 + 𝑓 𝑥 = 2 ∴ lim 𝑥→1 𝑓 𝑥 = 2 lim 𝑥→ −1 − 𝑓 𝑥 = 1 lim 𝑥→ −1 + 𝑓 𝑥 = 2 ∴ lim 𝑥→−1 𝑓 𝑥 = 𝐷𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 (𝐷𝑁𝐸)

1.1 – Limits: A Numerical and Graphical Approach Given the graph of a function, find the requested limits. lim 𝑥→ −3 − 𝑓 𝑥 = 3 lim 𝑥→ −3 + 𝑓 𝑥 = −1 lim 𝑥→−3 𝑓 𝑥 = 𝐷𝑁𝐸 𝑓 −3 = 1 lim 𝑥→ 0 − 𝑓 𝑥 = 2 lim 𝑥→ 0 + 𝑓 𝑥 = 2 lim 𝑥→0 𝑓 𝑥 = 𝑓 0 = 2 2 lim 𝑥→ 2 − 𝑓 𝑥 = 6 lim 𝑥→ 2 + 𝑓 𝑥 = −1 lim 𝑥→2 𝑓 𝑥 = 𝐷𝑁𝐸 𝑓 2 = 6 lim 𝑥→ 3 − 𝑓 𝑥 = 2 lim 𝑥→ 3 + 𝑓 𝑥 = −4 lim 𝑥→3 𝑓 𝑥 = 𝑓 3 = 𝐷𝑁𝐸 2

1.1 – Limits: A Numerical and Graphical Approach Given the graph of a function, find the requested limits. lim 𝑥→ −3 − 𝑓 𝑥 = −1 lim 𝑥→ −3 + 𝑓 𝑥 = −2 lim 𝑥→−3 𝑓 𝑥 = 𝐷𝑁𝐸 𝑓 −3 = −1 lim 𝑥→ 0 − 𝑓 𝑥 = 2 lim 𝑥→ 0 + 𝑓 𝑥 = 2 lim 𝑥→0 𝑓 𝑥 = 𝑓 0 = 2 2 lim 𝑥→ 1 − 𝑓 𝑥 = 1 lim 𝑥→ 1 + 𝑓 𝑥 = −3 lim 𝑥→1 𝑓 𝑥 = 𝐷𝑁𝐸 𝑓 1 = −3 lim 𝑥→ 2 − 𝑓 𝑥 = −1 lim 𝑥→ 2 + 𝑓 𝑥 = −3 lim 𝑥→2 𝑓 𝑥 = 𝑓 2 = 𝐷𝑁𝐸 −3

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Graphically lim 𝑥→∞ 𝑓 𝑥 = lim 𝑥→∞ 𝑔 𝑥 = ∞ ∞ lim 𝑥→−∞ 𝑓 𝑥 = lim 𝑥→−∞ 𝑔 𝑥 = ∞ −∞

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Graphically lim 𝑥→ 1 − 𝑓 𝑥 = lim 𝑥→∞ 𝑔 𝑥 = −∞ lim 𝑥→∞ 𝑓 𝑥 = lim 𝑥→ 0 − 𝑓 𝑥 = 1 −∞ lim 𝑥→−∞ 𝑔 𝑥 = lim 𝑥→ 1 + 𝑓 𝑥 = ∞ −1 lim 𝑥→−∞ 𝑓 𝑥 = lim 𝑥→ 0 + 𝑓 𝑥 = ∞

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Graphically 𝑓 𝑥 = 1 𝑥−1 lim 𝑥→∞ 1 𝑥−1 = lim 𝑥→−∞ 1 𝑥−1 = lim 𝑥→ 1 + 1 𝑥−1 = ∞ lim 𝑥→ 1 − 1 𝑥−1 = −∞ lim 𝑥→1 1 𝑥−1 = 𝐷𝑁𝐸

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 𝑥 = 1 𝑥−1 𝑓 10 = 1 10−1 =0.111 𝑓 100 = 1 100−1 =0.0101 𝑓 1000 = 1 1000−1 =0.001001 lim 𝑥→∞ 1 𝑥−1 =0

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 𝑥 = 1 𝑥−1 𝑓 −10 = 1 −10−1 =−0.0909 𝑓 −100 = 1 −100−1 =−0.0099 𝑓 −1000 = 1 −1000−1 =−0.00099 lim 𝑥→−∞ 1 𝑥−1 =0

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 𝑥 = 1 𝑥−1 𝑓 1.01 = 1 1.01−1 =100 𝑓 1.001 = 1 1.001−1 =1000 𝑓 1.0001 = 1 −1000−1 =10000 lim 𝑥→ 1 + 1 𝑥−1 =∞

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 𝑥 = 1 𝑥−1 𝑓 0.9 = 1 0.9−1 =−10 𝑓 .99 = 1 0.99−1 =−100 𝑓 0.999 = 1 0.999−1 =−1000 lim 𝑥→ 1 − 1 𝑥−1 =−∞

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Graphically 𝑓 𝑥 = −1 𝑥+4 +3 lim 𝑥→∞ −1 𝑥+4 +3= 3 lim 𝑥→−∞ −1 𝑥+4 +3= 3 lim 𝑥→ −4 − −1 𝑥+4 +3= ∞ lim 𝑥→− 4 + −1 𝑥+4 +3= −∞ lim 𝑥→4 −1 𝑥+4 +3= 𝐷𝑁𝐸

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 10 = −1 10+4 +3=2.92857 𝑓 𝑥 = −1 𝑥+4 +3 𝑓 100 = −1 100+4 +3=2.99038 𝑓 1000 = −1 1000+4 +3=2.99900 lim 𝑥→∞ −1 𝑥+4 +3=3

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 −10 = −1 −10+4 +3=3.16667 𝑓 𝑥 = −1 𝑥+4 +3 𝑓 −100 = −1 −100+4 +3=3.01042 𝑓 −1000 = −1 −1000+4 +3=3.00100 lim 𝑥→−∞ −1 𝑥+4 +3=3

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 −4.01 = −1 −4.01+4 +3=103 𝑓 𝑥 = −1 𝑥+4 +3 𝑓 −4.001 = −1 −4.001+4 +3=1003 𝑓 −4.0001 = −1 −4.0001+4 +3=10003 lim 𝑥→− 4 − −1 𝑥+4 +3=∞

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Infinity Numerically 𝑓 −3.9 = −1 −3.9+4 +3=−7 𝑓 𝑥 = −1 𝑥+4 +3 𝑓 −3.99 = −1 −3.99+4 +3=−97 𝑓 −3.999 = −1 −3.999+4 +3=−997 lim 𝑥→− 4 + −1 𝑥+4 +3=−∞

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Piecewise Functions

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Piecewise Functions lim 𝑥→3 𝑓 𝑥 = lim 𝑥→−2 𝑓 𝑥 = lim 𝑥→3 2𝑥−1= lim 𝑥→−2 − 𝑥 2 +4= lim 𝑥→3 2 3 −1= lim 𝑥→−2 − −2 2 +4= lim 𝑥→3 𝑓 𝑥 =5 lim 𝑥→−2 𝑓 𝑥 =0 𝑓 𝑥 = − 𝑥 2 +4 𝑥<1 2𝑥−1 𝑥≥1 lim 𝑥→1 𝑓 𝑥 = lim 𝑥→ 1 − 𝑓 𝑥 = lim 𝑥→ 1 + 𝑓 𝑥 = 1 − 𝑥 2 +4 2𝑥−1 lim 𝑥→ 1 − − 𝑥 2 +4= lim 𝑥→ 1 + 2𝑥−1= lim 𝑥→ 1 − 𝑓 𝑥 =3 lim 𝑥→ 1 + 𝑓 𝑥 =1 lim 𝑥→1 𝑓 𝑥 = 𝐷𝑁𝐸

1.1 – Limits: A Numerical and Graphical Approach Limits Involving Piecewise Functions lim 𝑥→−2 𝑓 𝑥 = lim 𝑥→ −2 − 𝑓 𝑥 = lim 𝑥→ −2 + 𝑓 𝑥 = lim 𝑥→− 2 − 𝑥+3= lim 𝑥→ −2 + 𝑥 2 = lim 𝑥→− 2 − −2+3= lim 𝑥→ −2 + −2 2 = lim 𝑥→ −2 + 𝑓 𝑥 =4 lim 𝑥→− 2 − 𝑓 𝑥 =1 lim 𝑥→−2 𝑓 𝑥 = 𝐷𝑁𝐸 𝑓 𝑥 = 𝑥+3 𝑥<−2 𝑥 2 −2≤𝑥≤1 −𝑥+2 𝑥>1 lim 𝑥→1 𝑓 𝑥 = lim 𝑥→ 1 − 𝑓 𝑥 = lim 𝑥→ 1 + 𝑓 𝑥 = lim 𝑥→ 1 − 𝑥 2 = lim 𝑥→− 1 + −𝑥+2= −2 1 𝑥+3 𝑥 2 −𝑥+2 lim 𝑥→ 1 − 1 2 = lim 𝑥→ 1 + − 1 +2= lim 𝑥→ 1 − 𝑓 𝑥 =1 lim 𝑥→ 1 + 𝑓 𝑥 =1 lim 𝑥→1 𝑓 𝑥 = 1

1.1 – Limits: A Numerical and Graphical Approach