Calculus Examples

$\frac{lim_{h \to 0} \tan (\frac{π}{4} + h) - 1}{h}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{h \to 0} \tan (\frac{π}{4} + h) - lim_{h \to 0} 1}{h}$

Step 1.2

Move the limit inside the trig function because tangent is continuous.

$\frac{\tan (lim_{h \to 0} \frac{π}{4} + h) - lim_{h \to 0} 1}{h}$

Step 1.3

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

$\frac{\tan (lim_{h \to 0} \frac{π}{4} + lim_{h \to 0} h) - lim_{h \to 0} 1}{h}$

Step 1.4

Evaluate the limit of $\frac{π}{4}$ which is constant as $h$ approaches $0$ .

$\frac{\tan (\frac{π}{4} + lim_{h \to 0} h) - lim_{h \to 0} 1}{h}$

Step 1.5

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

$\frac{\tan (\frac{π}{4} + lim_{h \to 0} h) - 1 \cdot 1}{h}$

Step 2

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

$\frac{\tan (\frac{π}{4} + 0) - 1 \cdot 1}{h}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $\frac{π}{4}$ and $0$ .

$\frac{\tan (\frac{π}{4}) - 1 \cdot 1}{h}$

Step 3.1.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{1 - 1 \cdot 1}{h}$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{h}$

Step 3.1.4

Subtract $1$ from $1$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$