Calculus Examples

$lim_{t \to \frac{3}{2}} \frac{2 \sin (t)}{3 \cos (t)}$

Step 1

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $t$ .

$\frac{2}{3} lim_{t \to \frac{3}{2}} \frac{\sin (t)}{\cos (t)}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $\frac{3}{2}$ .

$\frac{2}{3} \cdot \frac{lim_{t \to \frac{3}{2}} \sin (t)}{lim_{t \to \frac{3}{2}} \cos (t)}$

Step 3

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

$\frac{2}{3} \cdot \frac{\sin (lim_{t \to \frac{3}{2}} t)}{lim_{t \to \frac{3}{2}} \cos (t)}$

Step 4

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

$\frac{2}{3} \cdot \frac{\sin (lim_{t \to \frac{3}{2}} t)}{\cos (lim_{t \to \frac{3}{2}} t)}$

Step 5

Evaluate the limits by plugging in $\frac{3}{2}$ for all occurrences of $t$ .

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Step 5.1

Evaluate the limit of $t$ by plugging in $\frac{3}{2}$ for $t$ .

$\frac{2}{3} \cdot \frac{\sin (\frac{3}{2})}{\cos (lim_{t \to \frac{3}{2}} t)}$

Step 5.2

Evaluate the limit of $t$ by plugging in $\frac{3}{2}$ for $t$ .

$\frac{2}{3} \cdot \frac{\sin (\frac{3}{2})}{\cos (\frac{3}{2})}$

Step 6

Simplify the answer.

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Step 6.1

Convert from $\frac{\sin (\frac{3}{2})}{\cos (\frac{3}{2})}$ to $\tan (\frac{3}{2})$ .

$\frac{2}{3} \tan (\frac{3}{2})$

Step 6.2

Evaluate $\tan (\frac{3}{2})$ .

$\frac{2}{3} \cdot 14.10141994$

Step 6.3

Multiply $\frac{2}{3} \cdot 14.10141994$ .

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Step 6.3.1

Combine $\frac{2}{3}$ and $14.10141994$ .

$\frac{2 \cdot 14.10141994}{3}$

Step 6.3.2

Multiply $2$ by $14.10141994$ .

$\frac{28.20283989}{3}$

Step 6.4

Divide $28.20283989$ by $3$ .

$9.40094663$