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Math 53 1st Semester, Exams of Mathematics

University of the Philippines Baguio (UPB)Mathematics

Let f be the function whose graph isshown in the figure below.

Typology: Exams

2020/2021

Uploaded on 11/02/2022

xavier-domingo 🇵🇭

1 document

1 / 5

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Don't miss anything!

Math 53 1st Semester 2021-2022

Homework 2

Student Name/Number: Xavier D. Domingo

_____________________________________________________________________________________

Do as indicated and work independently. Upload your neat and complete solutions to our

Google Classroom on or before September 29, 2021 (11:59 pm). Submit your output in a

pdf format or in a zip file containing your photographed or scanned solutions. Name your file as

M53BF3_HW2_LastNameFirstName.pdf or M53BF3_HW2_LastNameFirstName.zip. Example:

M53BF3_HW2_PinoRodney.pdf

_____________________________________________________________________________________

Let

f

be the function whose graph is shown in the figure below.

(1.1)

Evaluate

f

(

−

1)

,

f

(0)

,

f

(2)

,

and

f

(3).

1.

f (-1) = Undefined

2.

f (0) = 1

3.

f (2) = Undefined

4.

f (3) = 0

(1.2)

Evaluate also

lim

𝑥→−1

𝑓(𝑥),lim

𝑥→0

𝑓(𝑥),lim

𝑥→2

𝑓(𝑥),lim

𝑥→3

𝑓(𝑥), lim

𝑥→−1

−

𝑓(𝑥), lim

𝑥→−1

𝑓(𝑥), lim

𝑥→0

−

𝑓(𝑥), lim

𝑥→−1

𝑓(𝑥), lim

𝑥→−2

−

𝑓(𝑥),

𝑎𝑛𝑑 lim

𝑥→3

+

𝑓(𝑥)

Partial preview of the text

Download Math 53 1st Semester and more Exams Mathematics in PDF only on Docsity!

Math 53 1st Semester 2021 - 2022

Homework 2

Student Name/Number: Xavier D. Domingo

Do as indicated and work independently. Upload your neat and complete solutions to our

Google Classroom on or before September 29, 2021 (11:59 pm). Submit your output in a

pdf format or in a zip file containing your photographed or scanned solutions. Name your file as

M53BF3_HW2_LastNameFirstName.pdf or M53BF3_HW2_LastNameFirstName.zip. Example:

M53BF3_HW2_PinoRodney.pdf

Let f be the function whose graph is shown in the figure below.

(1. 1 ) Evaluate f ( − 1) , f (0) , f (2) , and f (3).

f (-1) = Undefined
f (0) = 1
f (2) = Undefined
f (3) = 0

(1.2) Evaluate also

lim

𝑥→− 1

𝑓

( 𝑥

) , lim

𝑥→ 0

𝑓

( 𝑥

) , lim

𝑥→ 2

𝑓

( 𝑥

) , lim

𝑥→ 3

𝑓

( 𝑥

) , lim

𝑥→− 1

−

𝑓

( 𝑥

) , lim

𝑥→− 1

𝑓

( 𝑥

) , lim

𝑥→ 0

−

𝑓

( 𝑥

) , lim

𝑥→− 1

𝑓

( 𝑥

) , lim

𝑥→− 2

−

𝑓

( 𝑥

) ,

𝑎𝑛𝑑 lim

𝑥→ 3

𝑓(𝑥)

(a) lim

𝑥→− 1

(b) lim

𝑥→ 0

= DNE

(c) lim

𝑥→ 2

(d) lim

𝑥→ 3

= DNE

(e) lim

𝑥→− 1

−

(f) lim

𝑥→ 0

−

(g) lim

𝑥→ 2

−

(h) lim

𝑥→ 3

( 2 .) Prove using the definition of limits: lim

𝑥→ 3

*We want to find δ so that when |x-3|<8, |3x- 1 - 8|< ε

|3x- 1 - 8|< ε

|3x-9|< ε

|3(x-3)| < ε

3|x-3|< ε

|x- 3 |<

𝜀

3

Delta: |x-3|<

δ =

𝜀

3

Proof:

Suppose ε>0 is given

Choose δ =

𝜀

3

Assume |x-3|<

𝜀

3

Now, |3x- 1 - 8|

=|3x- 9 |

(c) lim

𝑥→ 18

√𝑥− 2 − 4

𝑡− 18

(√𝑥− 2 − 4 )(√𝑥− 2 + 4 )

( 𝑡− 18

) ( √

𝑥− 2 + 4 )

𝑡− 2 + 4 √

𝑡− 2 − 4 √

𝑡− 2 − 16

𝑡− 18 ( √

𝑡− 2 + 4 )

𝑡− 2 − 16

𝑡− 18 ( √

𝑡− 2 + 4 )

𝑡− 18

𝑡− 18 (√𝑡− 2 + 4 )

1

√

𝑡− 2 + 4

1

√ 18 − 2 + 4

1

√

16 + 4

1

4 + 4

1

8

(d) lim

𝑥→

1

3

−

[3x + 1 ] = [ 3 ( 0. 3 ) + 1 ]

= [ 0. 9 + 1 ]

= [ 1. 9 ]

(e) lim

𝑥→

1

3

𝑥+ 1

1 − 3 𝑥

( 1 / 3 )+ 1

1 − 3

( 1 / 3

)

( 1 / 3 + 1 )

1 - 3 ( 1 / 3 )

4 / 3

( 1 − 1

)( 3

)

4

0 ( 3 )

4 / 3

0

−

PS: Sir, nag direct substitution po ako sa item na ito gamit ang fraction kasi

po nalilito po ako kapag decimal number po ginamit ko sir. Nilagyan ko din

po ng graph to prove po kasi di po ako masyadong sure sa answer ko.

(d) lim

𝑥→+∞

𝑥+ 1

1 − 3 𝑥

( 𝑥+ 1

) (

1

𝑥

)

( 1 − 3 𝑥

) (

1

𝑥

)

1 +

1

𝑥

1

𝑥

− 3

1 + 0

0 − 3

Math 53 1st Semester, Exams of Mathematics

Related documents

Partial preview of the text

Download Math 53 1st Semester and more Exams Mathematics in PDF only on Docsity!

= DNE

= DNE

(c) lim

= [ 0. 9 + 1 ]

= [ 1. 9 ]

(e) lim

(d) lim