Calculus Examples

$lim_{x \to 1} x^{2} \sin (π x^{2})$

Step 1

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

$lim_{x \to 1} x^{2} \cdot lim_{x \to 1} \sin (π x^{2})$

Step 2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 1} x)}^{2} \cdot lim_{x \to 1} \sin (π x^{2})$

Step 3

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

${(lim_{x \to 1} x)}^{2} \cdot \sin (lim_{x \to 1} π x^{2})$

Step 4

Move the term $π$ outside of the limit because it is constant with respect to $x$ .

${(lim_{x \to 1} x)}^{2} \cdot \sin (π lim_{x \to 1} x^{2})$

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 1} x)}^{2} \cdot \sin (π {(lim_{x \to 1} x)}^{2})$

Step 6

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 6.1

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

$1^{2} \cdot \sin (π {(lim_{x \to 1} x)}^{2})$

Step 6.2

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

$1^{2} \cdot \sin (π \cdot 1^{2})$

Step 7

Simplify the answer.

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Step 7.1

One to any power is one.

$1 \cdot \sin (π \cdot 1^{2})$

Step 7.2

Multiply $\sin (π \cdot 1^{2})$ by $1$ .

$\sin (π \cdot 1^{2})$

Step 7.3

One to any power is one.

$\sin (π \cdot 1)$

Step 7.4

Multiply $π$ by $1$ .

$\sin (π)$

Step 7.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin (0)$

Step 7.6

The exact value of $\sin (0)$ is $0$ .

$0$