Question

Find a reasonable estimate of the limit

Answer 1

The correct option is c.

Explanation:

The given limit is

`lim_(x\rightarrow 2)(x^5-32)/(x^3-8)`

It is can be written as

`lim_(x\rightarrow 2)(x^5-2^5)/(x^3-2^3)`

According to the property of limits,

`lim_(x\rightarrow a)(x^n-a^n)/(x^m-a^m)=(n)/(m)(a)^(n-m)`

In the given limit, a=2, n=5 and m=3. Using the above property of limits we get

`lim_(x\rightarrow 2)(x^5-2^5)/(x^3-2^3)=(5)/(3)(2)^(5-3)`

`lim_(x\rightarrow 2)(x^5-2^5)/(x^3-2^3)=(5)/(3)(2)^(2)`

`lim_(x\rightarrow 2)(x^5-2^5)/(x^3-2^3)=(5)/(3)(4)`

`lim_(x\rightarrow 2)(x^5-2^5)/(x^3-2^3)=(20)/(3)`

`lim_(x\rightarrow 2)(x^5-2^5)/(x^3-2^3)=6.6667`

Therefore the correct option is c.

Answer 2

The answer is (c) ⇒ the value is 6.6667

Explanation:

∵
`\lim_(x\to \2) _2(x^(5)-32)/(x^(3)-8)`

∵ 32 = 2^5 , 8 = 2³

∴
`\lim_(x \to \2)_2 (x^(5)-2^(5))/(x^(3)-2^(3) )`

* by using the rule:

`\lim_(x\to\a)_a (x^(n)-a^(n))/(x^(m)-a^(m))=(n)/(m)(a)^(n-m)`

∴
`(5)/(3)(2)^(5-3)=(5)/(3)(2)^(2)=(20)/(3)`

∴ 20/3 = 6.6667 ⇒ answer (c)