Calculus Examples

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

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 $x$ approaches $\frac{π}{2}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{2}$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\sin^{2} (\frac{π}{2}) - 1 \cdot 1$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$1^{2} - 1 \cdot 1$

Step 3.1.2

One to any power is one.

$1 - 1 \cdot 1$

Step 3.1.3

Multiply $- 1$ by $1$ .

$1 - 1$

Step 3.2

Subtract $1$ from $1$ .

$0$