Calculus Examples

$lim_{x \to 1} \frac{5 x^{4} + 3^{x}}{\sqrt{x}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 5 x^{4} + 3^{x}}{lim_{x \to 1} \sqrt{x}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 5 x^{4} + lim_{x \to 1} 3^{x}}{lim_{x \to 1} \sqrt{x}}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 1} x^{4} + lim_{x \to 1} 3^{x}}{lim_{x \to 1} \sqrt{x}}$

Step 4

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{5 {(lim_{x \to 1} x)}^{4} + lim_{x \to 1} 3^{x}}{lim_{x \to 1} \sqrt{x}}$

Step 5

Move the limit into the exponent.

$\frac{5 {(lim_{x \to 1} x)}^{4} + 3^{lim_{x \to 1} x}}{lim_{x \to 1} \sqrt{x}}$

Step 6

Move the limit under the radical sign.

$\frac{5 {(lim_{x \to 1} x)}^{4} + 3^{lim_{x \to 1} x}}{\sqrt{lim_{x \to 1} x}}$

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{5 \cdot 1^{4} + 3^{lim_{x \to 1} x}}{\sqrt{lim_{x \to 1} x}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{5 \cdot 1^{4} + 3^{1}}{\sqrt{lim_{x \to 1} x}}$

Step 7.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{5 \cdot 1^{4} + 3^{1}}{\sqrt{1}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

One to any power is one.

$\frac{5 \cdot 1 + 3^{1}}{\sqrt{1}}$

Step 8.1.2

Multiply $5$ by $1$ .

$\frac{5 + 3^{1}}{\sqrt{1}}$

Step 8.1.3

Evaluate the exponent.

$\frac{5 + 3}{\sqrt{1}}$

Step 8.1.4

Add $5$ and $3$ .

$\frac{8}{\sqrt{1}}$

Step 8.2

Any root of $1$ is $1$ .

$\frac{8}{1}$

Step 8.3

Divide $8$ by $1$ .

$8$