Calculus Examples

$lim_{x \to 1} 2^{\frac{x - 7}{3}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit into the exponent.

$2^{lim_{x \to 1} \frac{x - 7}{3}}$

Step 1.2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$2^{\frac{1}{3} lim_{x \to 1} x - 7}$

Step 1.3

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

$2^{\frac{1}{3} (lim_{x \to 1} x - lim_{x \to 1} 7)}$

Step 1.4

Evaluate the limit of $7$ which is constant as $x$ approaches $1$ .

$2^{\frac{1}{3} (lim_{x \to 1} x - 1 \cdot 7)}$

Step 2

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

$2^{\frac{1}{3} (1 - 1 \cdot 7)}$

Step 3

Simplify the answer.

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Step 3.1

Multiply $- 1$ by $7$ .

$2^{\frac{1}{3} (1 - 7)}$

Step 3.2

Subtract $7$ from $1$ .

$2^{\frac{1}{3} \cdot - 6}$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Factor $3$ out of $- 6$ .

$2^{\frac{1}{3} \cdot (3 (- 2))}$

Step 3.3.2

Cancel the common factor.

$2^{\frac{1}{3} \cdot (3 \cdot - 2)}$

Step 3.3.3

Rewrite the expression.

$2^{- 2}$

Step 3.4

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

$\frac{1}{2^{2}}$

Step 3.5

Raise $2$ to the power of $2$ .

$\frac{1}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$