7.Limits Exercise 7.1

Evaluate the following limit:

1)`lim_(z -> -3) [sqrt("z" + 6)/"z"]`

`lim_(z -> -3) sqrt("z" + 6)/"z"`

= `(lim_(z -> - 3) sqrt(z + 6))/(lim_(z -> - 3) "z")   `

= `sqrt(-3 + 6)/-3`

= `sqrt(3)/-3`

= `-1/sqrt(3)`

Evaluate the following limit:

2)`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`

`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`

= `lim_(y -> -3) (((y^5 + 243)/(y + 3)))/(((y^3 + 27)/(y + 3)))   `

= `(lim_(y -> -3)[(y^5 - (- 3)^5)/(y - ( - 3))])/(lim_(y -> - 3)[(y^3 - ( - 3)^3)/(y - (- 3))]`

Evaluate the following limit:

3)`lim_(z -> -5)[((1/z + 1/5))/(z + 5)]`

`lim_(z -> -5)[(1/z + 1/5)/(z + 5)]`

= `lim_(z -> -5) (((5 + z)/(5z)))/(z + 5)`

= `lim_(z -> - 5) 1/(5z)`    ...[∵ z → – 5, z ≠ – 5 ∴ z + 5 ≠ 0]

= `(lim_(z -> - 5)(1))/(lim_(z -> - 5) (5z))`

= `1/(5( - 5))`

= `-1/25`

Evaluate the following limit:

4)`lim_(x -> 3)[sqrt(2x + 6)/x]`

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= `sqrt(2(3) + 6)/3`

= `sqrt(12)/3`

= `(2sqrt(3))/3`

Evaluate the following limit:

5)`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`

`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`

`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`

= (– 3) . (2)^–4     ...`[because  lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`

= `-3xx1/2^4`

= `-3/16`