Calculus Examples

$lim_{x \to 1.5} \frac{\sin (x - 2)}{x^{2} - 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1.5$ .

$\frac{lim_{x \to 1.5} \sin (x - 2)}{lim_{x \to 1.5} x^{2} - 4}$

Step 2

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

$\frac{\sin (lim_{x \to 1.5} x - 2)}{lim_{x \to 1.5} x^{2} - 4}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1.5$ .

$\frac{\sin (lim_{x \to 1.5} x - lim_{x \to 1.5} 2)}{lim_{x \to 1.5} x^{2} - 4}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $1.5$ .

$\frac{\sin (lim_{x \to 1.5} x - 1 \cdot 2)}{lim_{x \to 1.5} x^{2} - 4}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1.5$ .

$\frac{\sin (lim_{x \to 1.5} x - 1 \cdot 2)}{lim_{x \to 1.5} x^{2} - lim_{x \to 1.5} 4}$

Step 6

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

$\frac{\sin (lim_{x \to 1.5} x - 1 \cdot 2)}{{(lim_{x \to 1.5} x)}^{2} - lim_{x \to 1.5} 4}$

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $1.5$ .

$\frac{\sin (lim_{x \to 1.5} x - 1 \cdot 2)}{{(lim_{x \to 1.5} x)}^{2} - 1 \cdot 4}$

Step 8

Evaluate the limits by plugging in $1.5$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $1.5$ for $x$ .

$\frac{\sin (1.5 - 1 \cdot 2)}{{(lim_{x \to 1.5} x)}^{2} - 1 \cdot 4}$

Step 8.2

Evaluate the limit of $x$ by plugging in $1.5$ for $x$ .

$\frac{\sin (1.5 - 1 \cdot 2)}{{1.5}^{2} - 1 \cdot 4}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 1$ by $2$ .

$\frac{\sin (1.5 - 2)}{{1.5}^{2} - 1 \cdot 4}$

Step 9.1.2

Subtract $2$ from $1.5$ .

$\frac{\sin (- 0.5)}{{1.5}^{2} - 1 \cdot 4}$

Step 9.1.3

Evaluate $\sin (- 0.5)$ .

$\frac{- 0.47942553}{{1.5}^{2} - 1 \cdot 4}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Raise $1.5$ to the power of $2$ .

$\frac{- 0.47942553}{2.25 - 1 \cdot 4}$

Step 9.2.2

Multiply $- 1$ by $4$ .

$\frac{- 0.47942553}{2.25 - 4}$

Step 9.2.3

Subtract $4$ from $2.25$ .

$\frac{- 0.47942553}{- 1.75}$

Step 9.3

Divide $- 0.47942553$ by $- 1.75$ .

$0.27395745$