Solved on Feb 10, 2024

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

STEP 1

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

STEP 2

Recognize that as $x$ 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 $5^{x}$ and $4^{x}$ respectively.

STEP 4

Divide every term in the numerator and the denominator by $5^{x}$ , the highest base in the numerator, to simplify the limit.
$\lim _{x \rightarrow \infty} \frac{4^{x}-5^{x}}{3^{x}-4^{x}} = \lim _{x \rightarrow \infty} \frac{\frac{4^{x}}{5^{x}}-\frac{5^{x}}{5^{x}}}{\frac{3^{x}}{5^{x}}-\frac{4^{x}}{5^{x}}}$

STEP 5

Simplify the expression by applying the laws of exponents $a^{m}/a^{n} = a^{m-n}$ .
$\lim _{x \rightarrow \infty} \frac{\left(\frac{4}{5}\right)^{x}-1}{\left(\frac{3}{5}\right)^{x}-\left(\frac{4}{5}\right)^{x}}$

STEP 6

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

STEP 7

Apply the limits to the simplified expression.
$\lim _{x \rightarrow \infty} \frac{\left(\frac{4}{5}\right)^{x}-1}{\left(\frac{3}{5}\right)^{x}-\left(\frac{4}{5}\right)^{x}} = \frac{0-1}{0-0}$

STEP 8

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

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

STEP 1

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

STEP 2

Recognize that as $x$ 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 $5^{x}$ and $4^{x}$ respectively.

STEP 4

STEP 5

Simplify the expression by applying the laws of exponents $a^{m}/a^{n} = a^{m-n}$ .
$\lim _{x \rightarrow \infty} \frac{\left(\frac{4}{5}\right)^{x}-1}{\left(\frac{3}{5}\right)^{x}-\left(\frac{4}{5}\right)^{x}}$

STEP 6

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

STEP 7

Apply the limits to the simplified expression.
$\lim _{x \rightarrow \infty} \frac{\left(\frac{4}{5}\right)^{x}-1}{\left(\frac{3}{5}\right)^{x}-\left(\frac{4}{5}\right)^{x}} = \frac{0-1}{0-0}$

STEP 8

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

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

STEP 1

Assumptions1. We are evaluating the limit of a function as x x x approaches infinity.2. The function is a quotient of two expressions: 4x−5x 4^{x} - 5^{x} 4x−5x in the numerator and 3x−4x 3^{x} - 4^{x} 3x−4x in the denominator.

STEP 2

Recognize that as x 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} 5x and 4x 4^{x} 4x respectively.

STEP 4

STEP 5

STEP 6

As x x x approaches infinity, (45)x \left(\frac{4}{5}\right)^{x} (54​)x and (35)x \left(\frac{3}{5}\right)^{x} (53​)x approach 0 because 45 \frac{4}{5} 54​ and 35 \frac{3}{5} 53​ 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} x→∞lim​(53​)x−(54​)x(54​)x−1​=0−00−1​

STEP 8

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

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

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

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

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

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

Apply the limits to the simplified expression.
$\lim _{x \rightarrow \infty} \frac{\left(\frac{4}{5}\right)^{x}-1}{\left(\frac{3}{5}\right)^{x}-\left(\frac{4}{5}\right)^{x}} = \frac{0-1}{0-0}$

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

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.

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 $-\infty$ .