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Question Find the limit of the expression (4x−5x)/(3x−4x)(4^x - 5^x) / (3^x - 4^x) as xx approaches infinity.

Studdy Solution

STEP 1

Assumptions
1. We are evaluating the limit of a function as x x approaches infinity.
2. The function is a quotient of two expressions: 4x−5x 4^{x} - 5^{x} in the numerator and 3x−4x 3^{x} - 4^{x} in the denominator.

STEP 2

Recognize that as x x approaches infinity, the terms with the highest base in the exponent will dominate the growth of the function.

STEP 3

In both the numerator and the denominator, the term with the highest base is 5x 5^{x} and 4x 4^{x} respectively.

STEP 4

Divide every term in the numerator and the denominator by 5x 5^{x} , the highest base in the numerator, to simplify the limit.
lim⁡x→∞4x−5x3x−4x=lim⁡x→∞4x5x−5x5x3x5x−4x5x \lim _{x \rightarrow \infty} \frac{4^{x}-5^{x}}{3^{x}-4^{x}} = \lim _{x \rightarrow \infty} \frac{\frac{4^{x}}{5^{x}}-\frac{5^{x}}{5^{x}}}{\frac{3^{x}}{5^{x}}-\frac{4^{x}}{5^{x}}}

STEP 5

Simplify the expression by applying the laws of exponents am/an=am−n a^{m}/a^{n} = a^{m-n} .
lim⁡x→∞(45)x−1(35)x−(45)x \lim _{x \rightarrow \infty} \frac{\left(\frac{4}{5}\right)^{x}-1}{\left(\frac{3}{5}\right)^{x}-\left(\frac{4}{5}\right)^{x}}

STEP 6

As x x approaches infinity, (45)x \left(\frac{4}{5}\right)^{x} and (35)x \left(\frac{3}{5}\right)^{x} approach 0 because 45 \frac{4}{5} and 35 \frac{3}{5} are fractions less than 1.

STEP 7

Apply the limits to the simplified expression.
lim⁡x→∞(45)x−1(35)x−(45)x=0−10−0 \lim _{x \rightarrow \infty} \frac{\left(\frac{4}{5}\right)^{x}-1}{\left(\frac{3}{5}\right)^{x}-\left(\frac{4}{5}\right)^{x}} = \frac{0-1}{0-0}

STEP 8

Simplify the expression after applying the limits.
−10 \frac{-1}{0}

STEP 9

Recognize that division by zero is undefined, but in the context of limits, if the numerator is non-zero and the denominator approaches zero, the limit approaches ±∞ \pm\infty , depending on the signs of the numerator and denominator.

STEP 10

Since the numerator is negative and the denominator is approaching zero from the negative side, the limit is −∞ -\infty .
The solution to the limit is −∞ -\infty .

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