Calculus Examples

$lim_{z \to 2} \frac{\csc (z - 3)}{z}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $z$ approaches $2$ .

$\frac{lim_{z \to 2} \csc (z - 3)}{lim_{z \to 2} z}$

Step 2

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

$\frac{\csc (lim_{z \to 2} z - 3)}{lim_{z \to 2} z}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $2$ .

$\frac{\csc (lim_{z \to 2} z - lim_{z \to 2} 3)}{lim_{z \to 2} z}$

Step 4

Evaluate the limit of $3$ which is constant as $z$ approaches $2$ .

$\frac{\csc (lim_{z \to 2} z - 1 \cdot 3)}{lim_{z \to 2} z}$

Step 5

Evaluate the limits by plugging in $2$ for all occurrences of $z$ .

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Step 5.1

Evaluate the limit of $z$ by plugging in $2$ for $z$ .

$\frac{\csc (2 - 1 \cdot 3)}{lim_{z \to 2} z}$

Step 5.2

Evaluate the limit of $z$ by plugging in $2$ for $z$ .

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 1$ by $3$ .

$\frac{\csc (2 - 3)}{2}$

Step 6.1.2

Subtract $3$ from $2$ .

$\frac{\csc (- 1)}{2}$

Step 6.1.3

Evaluate $\csc (- 1)$ .

$\frac{- 57.29868849}{2}$

Step 6.2

Divide $- 57.29868849$ by $2$ .

$- 28.64934424$