if h≠ 0, then (1/(x+h+1)-1/(x+1))/h=
please help me to understand this question and how to answer it.
[ 1 / ( x + h + 1 ) - 1 / ( x + 1 ) ] / h =
[ x + 1 - ( x + h + 1 ) ) / ( ( x + h + 1 ) ∙ ( x + 1 ) ] / h =
[ x + 1 - x - h - 1 ) / ( ( x + h + 1 ) ∙ ( x + 1 ) ] / h =
- h / [ ( x + h + 1 ) ∙ ( x + 1 ) ] / h =
- h / [ h ∙ ( x + h + 1 ) ∙ ( x + 1 ) ] =
- 1 / [ ( x + h + 1 ) ∙ ( x + 1 ) ] =
- 1 / ( x ∙ x + x ∙ h + 1 ∙ x + 1 ∙ x + 1 ∙ h + 1 ∙ 1 ) =
- 1 / ( x² + x ∙ h + x + x + h + 1 ) =
- 1 / ( x² + 2 x + x ∙ h + h + 1 )
[ 1 / ( x + h + 1 ) - 1 / ( x + 1 ) ] / h = - 1 / ( x² + 2 x + x ∙ h + h + 1 )
If your question mean:
lim h→ 0 ( 1 / ( x + h + 1 ) - 1 / ( x + 1 ) ) / h
then:
lim h→ 0 [ - 1 / ( x² + 2 x + x ∙ h + h + 1 ) ] =
- 1 / ( x² + 2 x + x ∙ 0 + 0 + 1 ) =
- 1 / ( x² + 2 x + 0 + 0 + 1 ) =
- 1 / ( x² + 2 x + 1 ) = ,
- 1 / ( x + 1 )²
If your question mean:
lim h→∞ [ - 1 / ( x² + 2 x + x ∙ h + h + 1 ) ]
then:
lim h→ ∞ [ - 1 / ( ∞ + 2 ∙ ∞ + ∞ ∙ h + h + 1 ) ] = - 1 / ∞ = 0