Calculus Examples

$lim_{t \to 0^{-}} (\sin^{2} (\frac{1}{t})) \cdot 3^{\frac{1}{t}}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $0$ .

$lim_{t \to 0^{-}} \sin^{2} (\frac{1}{t}) \cdot lim_{t \to 0^{-}} 3^{\frac{1}{t}}$

Step 1.2

Move the exponent $2$ from $\sin^{2} (\frac{1}{t})$ outside the limit using the Limits Power Rule.

${(lim_{t \to 0^{-}} \sin (\frac{1}{t}))}^{2} \cdot lim_{t \to 0^{-}} 3^{\frac{1}{t}}$

Step 2

Since the exponent $\frac{1}{t}$ approaches $- \infty$ , the quantity $3^{\frac{1}{t}}$ approaches $0$ .

${(lim_{t \to 0^{-}} \sin (\frac{1}{t}))}^{2} \cdot 0$

Step 3

Make a table to show the behavior of the function $\sin (\frac{1}{t})$ as $t$ approaches $0$ from the left.

$\begin{array}{c} t & \sin (\frac{1}{t}) \\ - 0.1 & 0.54402111 \\ - 0.01 & 0.50636564 \\ - 0.001 & - 0.99388865 \\ - 0.0001 & - 0.99388865 \end{array}$

Step 4

As the $t$ values approach $0$ , the function values approach $- 0.994$ . Thus, the limit of $\sin (\frac{1}{t})$ as $t$ approaches $0$ from the left is $- 0.994$ .

${(- 0.994)}^{2} \cdot 0$

Step 5

Simplify the answer.

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Step 5.1

Raise $- 0.994$ to the power of $2$ .

$0.988036 \cdot 0$

Step 5.2

Multiply $0.988036$ by $0$ .

$0$