Question

Use the limit theorem and the properties of limits to find the limit.
Picture provided below

Answer 1

Option B

Explanation:

We know that the limit when x tends to infinity of:

`(1)/(x ^ n)` is approximately zero when n is a positive real number. The term of greatest exponent in the function is
`x ^ 2`

Based on this we do the following.

Divide each term of the numerator and denominator between the term with the greatest exponent of the expression. Then it is:

`\lim_(x\to \infty)((x-3)(x+2))/(2x^2 + x +1)\\\\= \lim_(x\to \infty) ((x^2 -x -6))/(2x^2 + x +1)\\\\\\ \lim_(x \to \infty)((x^2)/(x^2) -(x)/(x^2) -(6)/(x^2))/(2(x^2)/(x^2) + (x)/(x^2) +(1)/(x^2))\\\\`

Then as the
`(1)/(\infty) \to 0` then we have left:

`= \lim_(x \to \infty) ((x^2)/(x^2))/(2(x^2)/(x^2))\\\\\lim_(x \to \infty)(1)/(2) = (1)/(2)`

The answer is: Option b

Answer 2

b. 1/2

Explanation:

lim (x -3)(x +2)

x-->-∞ ---------------

2x^2 + x +1

= lim (x^2 -3x +2x - 6)

x-->-∞ -----------------------

2x^2 + x +1

= lim (x^2 -x - 6)

x-->-∞ -----------------------

2x^2 + x +1

When we plug in x = -∞, we get indeterminate form.

Now we have to use the L'hospital rule.

d/dx (x^2 - x - 6) = 2x -1

d/dx (2x^2 + x + 1) = 4x + 1

Now apply the limit

lim (2x - 1) / (4x + 1)

x--->-∞

Here we have to degree of the numerator and the denominator of the same. In this case, if x --> -∞, we get the result as the coefficient of the leading term as the result.

According to the rule, we get

= 2/4

Which can simplified as 1/2

The answer is 1/2

Hope this will helpful.

Thank you.