Question

If a and b are positive constants, then limx→[infinity] ln(bx+1)/ln(ax2+3)=

A. 0
B. 1/2
C. 1/2ab
D. 2
E. Infinity

Answer 1

The limit of the given expression as x approaches infinity is 0.

Step-by-step explanation:

To find the limit of the given expression as x approaches infinity, we can use the properties of logarithms. We know that ln(bx + 1) = ln(b) + ln(x + 1) and ln(ax^2 + 3) = ln(a) + ln(x^2 + 3). So, our expression becomes:

lim(x->∞) ln(bx + 1)/ln(ax^2 + 3) = lim(x->∞) (ln(b) + ln(x + 1))/(ln(a) + ln(x^2 + 3)).

As x approaches infinity, the terms ln(x + 1) and ln(x^2 + 3) become much larger compared to ln(b) and ln(a), respectively. Therefore, they dominate the expression, and we can ignore the other terms. So, the limit becomes:

lim(x->∞) (ln(x + 1))/(ln(x^2 + 3)).

We can simplify this by using the fact that ln(x + 1) grows much slower than ln(x^2 + 3) as x approaches infinity. So, the limit is:

lim(x->∞) (ln(x + 1))/(ln(x^2 + 3)) = 0.

Answer 2

To find the limit of the given expression, apply the limit laws and properties of logarithms. The limit is equal to the constant ratio ln(b)/ln(a).

Step-by-step explanation:

To find the limit of the given expression, we can use the properties of logarithms and the limit laws. First, we rewrite the expression as ln(bx+1)/ln(ax^2+3) = ln(bx+1)/ln((ax^2+3)^{1/2}), using the property ln(x^a) = a ln(x). Next, we apply the limit laws:

• lim x->∞ ln(bx+1)/ln((ax^2+3)^{1/2}) = ln(b)/ln(a) [using the limit law lim x->∞ ln(x)/ln(x) = 1]
• ln(b)/ln(a) is a constant value, so the limit is equal to that constant.

Therefore, the limit of ln(bx+1)/ln(ax^2+3) as x approaches infinity is a constant value, which is the ratio ln(b)/ln(a). The correct answer is

Option C: 1/2ab

.