Calculus Examples

$lim_{x \to 0} \frac{9}{x^{x}} - \frac{5^{x}}{x}$

Step 1

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} \frac{9}{x^{x}} - \frac{5^{x}}{x}$

Step 2

Evaluate the limits by plugging in the value for the variable.

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Step 2.1

Evaluate the limit of $\frac{9}{x^{x}} - \frac{5^{x}}{x}$ by plugging in $0$ for $x$ .

$\frac{9}{0^{0}} - \frac{5^{0}}{0}$

Step 2.2

Since $\frac{9}{0^{0}} - \frac{5^{0}}{0}$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

$lim_{x \to 0^{+}} \frac{9}{x^{x}} - \frac{5^{x}}{x}$

Step 4

Evaluate the right-sided limit.

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Step 4.1

Evaluate the limit.

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Step 4.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0^{+}} \frac{9}{x^{x}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.1.2

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$9 lim_{x \to 0^{+}} \frac{1}{x^{x}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.1.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$9 \frac{lim_{x \to 0^{+}} 1}{lim_{x \to 0^{+}} x^{x}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$9 \frac{1}{lim_{x \to 0^{+}} x^{x}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.2

Use the properties of logarithms to simplify the limit.

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Step 4.2.1

Rewrite $x^{x}$ as $e^{\ln (x^{x})}$ .

$9 \frac{1}{lim_{x \to 0^{+}} e^{\ln (x^{x})}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.2.2

Expand $\ln (x^{x})$ by moving $x$ outside the logarithm.

$9 \frac{1}{lim_{x \to 0^{+}} e^{x \ln (x)}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.3

Move the limit into the exponent.

$9 \frac{1}{e^{lim_{x \to 0^{+}} x \ln (x)}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.4

Rewrite $x \ln (x)$ as $\frac{\ln (x)}{\frac{1}{x}}$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} \frac{\ln (x)}{\frac{1}{x}}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5

Apply L'Hospital's rule.

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Step 4.5.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 4.5.1.1

Take the limit of the numerator and the limit of the denominator.

$9 \frac{1}{e^{\frac{lim_{x \to 0^{+}} \ln (x)}{lim_{x \to 0^{+}} \frac{1}{x}}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.1.2

As $x$ approaches $0$ from the right side, $\ln (x)$ decreases without bound.

$9 \frac{1}{e^{\frac{- \infty}{lim_{x \to 0^{+}} \frac{1}{x}}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.1.3

Since the numerator is a constant and the denominator approaches $0$ when $x$ approaches $0$ from the right, the fraction $\frac{1}{x}$ approaches infinity.

$9 \frac{1}{e^{\frac{- \infty}{\infty}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.1.4

Infinity divided by infinity is undefined.

Undefined

$9 \frac{1}{e^{\frac{- \infty}{\infty}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.2

Since $\frac{- \infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0^{+}} \frac{\ln (x)}{\frac{1}{x}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [\frac{1}{x}]}$

Step 4.5.3

Find the derivative of the numerator and denominator.

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Step 4.5.3.1

Differentiate the numerator and denominator.

$9 \frac{1}{e^{lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [\frac{1}{x}]}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [\frac{1}{x}]}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.3.3

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [x^{- 1}]}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 1$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{- x^{- 2}}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.3.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{- \frac{1}{x^{2}}}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.4

Multiply the numerator by the reciprocal of the denominator.

$9 \frac{1}{e^{lim_{x \to 0^{+}} \frac{1}{x} \cdot - 1 x^{2}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.5

Combine $\frac{1}{x}$ and $x^{2}$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} - \frac{x^{2}}{x}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.5.6.1

Factor $x$ out of $x^{2}$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} - \frac{x \cdot x}{x}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.6.2

Cancel the common factors.

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Step 4.5.6.2.1

Raise $x$ to the power of $1$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} - \frac{x \cdot x}{x^{1}}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.6.2.2

Factor $x$ out of $x^{1}$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} - \frac{x \cdot x}{x \cdot 1}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.6.2.3

Cancel the common factor.

$9 \frac{1}{e^{lim_{x \to 0^{+}} - \frac{x \cdot x}{x \cdot 1}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.6.2.4

Rewrite the expression.

$9 \frac{1}{e^{lim_{x \to 0^{+}} - \frac{x}{1}}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.5.6.2.5

Divide $x$ by $1$ .

$9 \frac{1}{e^{lim_{x \to 0^{+}} - x}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.6

Evaluate the limit.

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Step 4.6.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$9 \frac{1}{e^{- lim_{x \to 0^{+}} x}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$9 \frac{1}{e^{0}} - lim_{x \to 0^{+}} \frac{5^{x}}{x}$

Step 4.7

Since the numerator is positive and the denominator $x$ approaches zero and is greater than zero for $x$ near $0$ to the right, the function increases without bound.

$9 \frac{1}{e^{0}} - \infty$

Step 4.8

Simplify the answer.

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Step 4.8.1

Simplify each term.

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Step 4.8.1.1

Anything raised to $0$ is $1$ .

$9 (\frac{1}{1}) - \infty$

Step 4.8.1.2

Cancel the common factor of $1$ .

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Step 4.8.1.2.1

Cancel the common factor.

$9 (\frac{1}{1}) - \infty$

Step 4.8.1.2.2

Rewrite the expression.

$9 \cdot 1 - \infty$

Step 4.8.1.3

Multiply $9$ by $1$ .

$9 - \infty$

Step 4.8.1.4

A non-zero constant times infinity is infinity.

$9 + - \infty$

Step 4.8.2

Infinity plus or minus a number is infinity.

$- \infty$

Step 5

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$