Roboguru

Jika f(x)=x2−2x−13 nilai dari h→0lim​hf(x+h)−f(x)​ adalah ....

Pertanyaan

Jika straight f left parenthesis straight x right parenthesis equals straight x squared minus 2 straight x minus 13 nilai dari limit as straight h rightwards arrow 0 of fraction numerator straight f open parentheses straight x plus straight h close parentheses minus straight f open parentheses straight x close parentheses over denominator straight h end fraction adalah ....

 

Pembahasan Soal:

Mencari f(x+h)

table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses straight x plus straight h close parentheses end cell equals cell open parentheses straight x plus straight h close parentheses squared minus 2 open parentheses straight x plus straight h close parentheses minus 13 end cell row blank equals cell open parentheses straight x plus straight h close parentheses open parentheses straight x plus straight h close parentheses minus 2 straight x minus 2 straight h minus 13 end cell row blank equals cell straight x squared plus xh plus xh plus straight h squared minus 2 straight x minus 2 straight h minus 13 end cell row blank equals cell straight x squared plus 2 xh plus straight h squared minus 2 straight x minus 2 straight h minus 13 end cell end table

Subsitusi ke nilai lim

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight h rightwards arrow 0 of fraction numerator straight f open parentheses straight x plus straight h close parentheses minus straight f open parentheses straight x close parentheses over denominator straight h end fraction end cell equals cell limit as h rightwards arrow 0 of fraction numerator open parentheses x squared plus 2 x h plus h squared minus 2 x minus 2 h minus 13 close parentheses minus open parentheses x squared minus 2 x minus 13 close parentheses over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator x squared plus 2 x h plus h squared minus 2 x minus 2 h minus 13 minus x squared plus 2 x plus 13 over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator 2 x h plus h squared minus 2 h over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of 2 x plus h minus 2 end cell row blank equals cell 2 x minus 2 end cell end table

Jadi hasil dari straight f left parenthesis straight x right parenthesis equals straight x squared minus 2 straight x minus 13 nilai dari limit as straight h rightwards arrow 0 of fraction numerator straight f open parentheses straight x plus straight h close parentheses minus straight f open parentheses straight x close parentheses over denominator straight h end fraction= 2x -2

 

 

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Mahasiswa/Alumni UIN Walisongo Semarang

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Tentukan hasil dari: b) x→−4lim​(x2−6)3

Pembahasan Soal:

Diketahui:

Limit fungsi berpangkat.

Substitusi variabel begin mathsize 14px style x end style dengan begin mathsize 14px style negative 4 end style pada fungsi seperti berikut:

Error converting from MathML to accessible text.

Maka, nilai dari Error converting from MathML to accessible text..

0

Roboguru

Jika diketahui x→clim​f(x)=5danx→clim​g(x)=8 Maka nilai dari x→clim​[f(x)]2−2g(x)​=...

Pembahasan Soal:

Ingat sifat-sifat limit berikut :

rightwards double arrow limit as x rightwards arrow c of straight k g left parenthesis x right parenthesis equals straight k limit as x rightwards arrow c of g left parenthesis x right parenthesis rightwards double arrow limit as x rightwards arrow c of open square brackets f left parenthesis x right parenthesis plus-or-minus g left parenthesis x right parenthesis close square brackets equals limit as x rightwards arrow c of f left parenthesis x right parenthesis plus-or-minus limit as x rightwards arrow c of g left parenthesis x right parenthesis rightwards double arrow limit as x rightwards arrow c of open square brackets f left parenthesis x right parenthesis close square brackets to the power of n equals open square brackets limit as x rightwards arrow c of f left parenthesis x right parenthesis close square brackets to the power of n

Berdasarkan sifat-sifat tersebut, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow c of square root of open square brackets f left parenthesis x right parenthesis close square brackets squared minus 2 g left parenthesis x right parenthesis end root end cell equals cell square root of limit as x rightwards arrow c of open parentheses open square brackets f left parenthesis x right parenthesis close square brackets squared minus 2 g left parenthesis x right parenthesis close parentheses end root end cell row blank equals cell square root of open square brackets limit as x rightwards arrow c of f left parenthesis x right parenthesis close square brackets squared minus 2 limit as x rightwards arrow c of g left parenthesis x right parenthesis end root end cell row blank equals cell square root of 5 squared minus 2 left parenthesis 8 right parenthesis end root end cell row blank equals cell square root of 9 end cell row blank equals 3 end table

Jadi, diperoleh nilai limit tersebut adalah 3.

0

Roboguru

Jika x→clim​f(x)=6danx→clim​g(x)=−8. Carilah setiap limit berikut. a. x→clim​f2(x)−g2(x)​

Pembahasan Soal:

Sifat-sifat limit :

1. begin mathsize 14px style limit as x rightwards arrow c of open parentheses f open parentheses x close parentheses plus-or-minus g open parentheses x close parentheses close parentheses equals limit as x rightwards arrow c of f open parentheses x close parentheses plus-or-minus limit as x rightwards arrow c of g open parentheses x close parentheses end style

2. begin mathsize 14px style limit as x rightwards arrow c of f to the power of n open parentheses x close parentheses equals open parentheses limit as x rightwards arrow c of f open parentheses x close parentheses close parentheses to the power of n end style

3. begin mathsize 14px style limit as x rightwards arrow c of n-th root of f open parentheses x close parentheses end root equals n-th root of limit as x rightwards arrow c of f open parentheses x close parentheses end root end style , asalkan begin mathsize 14px style limit as x rightwards arrow c of f open parentheses x close parentheses greater or equal than 0 comma space straight n space genap end style 

Dengan menggunakan sifat-sifat limit fungsi diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow c of square root of f squared left parenthesis x right parenthesis minus g squared left parenthesis x right parenthesis end root end cell equals cell square root of limit as x rightwards arrow c of space open parentheses f squared left parenthesis x right parenthesis minus g squared left parenthesis x right parenthesis close parentheses end root end cell row blank equals cell square root of limit as x rightwards arrow c of space f squared left parenthesis x right parenthesis minus limit as x rightwards arrow c of space g squared left parenthesis x right parenthesis end root end cell row blank equals cell square root of open parentheses limit as x rightwards arrow c of space f left parenthesis x right parenthesis close parentheses squared minus open parentheses limit as x rightwards arrow c of space g left parenthesis x right parenthesis close parentheses squared end root end cell row blank equals cell square root of 6 squared minus left parenthesis negative 8 right parenthesis squared end root end cell row blank equals cell square root of 36 minus 64 end root end cell row blank equals cell square root of negative 28 end root end cell row blank equals cell 2 square root of negative 7 end root end cell end table end style 

Dengan demikian, diperoleh begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow c of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of f squared left parenthesis x right parenthesis minus g squared left parenthesis x right parenthesis end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of negative 7 end root end cell end table end style.

0

Roboguru

Tuliskan dan tunjukkan sifat-sifat limit yang mana saja dapat digunakan untuk menyelesaikan limit fungsi berikut x→1lim​(2x−1)4!

Pembahasan Soal:

Sifat-sifat limit yang digunakan :

1. begin mathsize 14px style limit as x rightwards arrow a of open parentheses f left parenthesis x right parenthesis close parentheses to the power of n equals open parentheses limit as x rightwards arrow a of f left parenthesis x right parenthesis close parentheses to the power of n end style

2. begin mathsize 14px style limit as x rightwards arrow a of open parentheses f left parenthesis x right parenthesis plus-or-minus g left parenthesis x right parenthesis close parentheses equals limit as x rightwards arrow a of f left parenthesis x right parenthesis plus-or-minus limit as x rightwards arrow a of g left parenthesis x right parenthesis end style

3. begin mathsize 14px style limit as x rightwards arrow a of k times f left parenthesis x right parenthesis equals k times limit as x rightwards arrow a of f left parenthesis x right parenthesis end style

4. begin mathsize 14px style limit as x rightwards arrow a of x equals a end style

5. begin mathsize 14px style limit as x rightwards arrow a of k equals k end style

 

Jadi nilai

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of left parenthesis 2 x minus 1 right parenthesis to the power of 4 end cell equals cell left parenthesis limit as x rightwards arrow 1 of left parenthesis 2 x minus 1 right parenthesis right parenthesis to the power of 4 space end cell row blank equals cell left parenthesis limit as x rightwards arrow 1 of 2 x minus limit as x rightwards arrow 1 of 1 right parenthesis to the power of 4 space end cell row blank equals cell left parenthesis 2 times limit as x rightwards arrow 1 of x minus limit as x rightwards arrow 1 of 1 right parenthesis to the power of 4 space end cell row blank equals cell left parenthesis 2 times open parentheses 1 close parentheses minus 1 right parenthesis to the power of 4 end cell row blank equals cell left parenthesis 2 minus 1 right parenthesis to the power of 4 space end cell row blank equals cell 1 to the power of 4 end cell row blank equals 1 end table end style

0

Roboguru

x→1lim​(x2+4x−9)=…

Pembahasan Soal:

Substitusikan nilai x ke fungsi sehingga menjadi

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of space left parenthesis x squared plus 4 x minus 9 right parenthesis end cell equals cell 1 squared plus 4 times 1 minus 9 end cell row blank equals cell 1 plus 4 minus 9 end cell row blank equals cell negative 4 end cell end table end style   

Jadi, nilai limit dari fungsi tersebut adalah begin mathsize 14px style negative 4 end style.

0

Roboguru

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