evaluate the limit 5x^(ln5+1)/(ln8x+1)
it this all in the exponent?
5x(ln5+1)/(ln8x+1)
limit as x -> ??
To evaluate the limit of the given expression, let's start by rewriting it using logarithmic properties.
The expression can be written as: lim(x → 0) [5x^(ln5+1)] / [ln(8x)+1].
Now, let's determine the behavior of the expression as x approaches zero.
First, we notice that as x approaches zero, the term x^(ln5+1) will also approach zero since any positive number raised to a negative power tends to zero.
Next, let's consider the term ln(8x) as x approaches zero. ln(8x) can be rewritten as ln(8) + ln(x) using the logarithmic property that the logarithm of a product is the sum of the logarithms. As x approaches zero, ln(8) remains constant, and ln(x) approaches negative infinity, resulting in ln(8x) also approaching negative infinity.
Now, let's evaluate the limit:
lim(x → 0) [5x^(ln5+1)] / [ln(8x)+1].
As x approaches zero, the numerator approaches zero, and the denominator approaches negative infinity. When dealing with a limit where the denominator approaches negative infinity, we can multiply the numerator and denominator by the reciprocal of the denominator to simplify the expression.
lim(x → 0) [5x^(ln5+1)] / [ln(8x)+1] = lim(x → 0) [5x^(ln5+1)] * [1 / (ln(8x)+1)].
Now, the limit becomes:
lim(x → 0) [5x^(ln5+1)] * [1 / (ln(8x)+1)].
Next, we can substitute the limiting value of x, which is zero, into the expression:
[5(0)^(ln5+1)] * [1 / (ln(8(0))+1)] = 0 * [1 / (-∞ + 1)].
The term 0 multiplied by any finite number is zero. Therefore, the limit is equal to zero:
lim(x → 0) [5x^(ln5+1)] / [ln(8x)+1] = 0.