Calculus Examples

$lim_{x \to 8} \frac{7 x^{2} + x \cdot \sin (x)}{3 - x^{2} + \sin (7 x^{2})}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 7 x^{2} + x \cdot \sin (x)}{lim_{x \to 8} 3 - x^{2} + \sin (7 x^{2})}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 7 x^{2} + lim_{x \to 8} x \cdot \sin (x)}{lim_{x \to 8} 3 - x^{2} + \sin (7 x^{2})}$

Step 3

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 lim_{x \to 8} x^{2} + lim_{x \to 8} x \cdot \sin (x)}{lim_{x \to 8} 3 - x^{2} + \sin (7 x^{2})}$

Step 4

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

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (x)}{lim_{x \to 8} 3 - x^{2} + \sin (7 x^{2})}$

Step 5

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot lim_{x \to 8} \sin (x)}{lim_{x \to 8} 3 - x^{2} + \sin (7 x^{2})}$

Step 6

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

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (lim_{x \to 8} x)}{lim_{x \to 8} 3 - x^{2} + \sin (7 x^{2})}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (lim_{x \to 8} x)}{lim_{x \to 8} 3 - lim_{x \to 8} x^{2} + lim_{x \to 8} \sin (7 x^{2})}$

Step 8

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (lim_{x \to 8} x)}{3 - lim_{x \to 8} x^{2} + lim_{x \to 8} \sin (7 x^{2})}$

Step 9

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

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (lim_{x \to 8} x)}{3 - {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} \sin (7 x^{2})}$

Step 10

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

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (lim_{x \to 8} x)}{3 - {(lim_{x \to 8} x)}^{2} + \sin (lim_{x \to 8} 7 x^{2})}$

Step 11

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (lim_{x \to 8} x)}{3 - {(lim_{x \to 8} x)}^{2} + \sin (7 lim_{x \to 8} x^{2})}$

Step 12

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

$\frac{7 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x \cdot \sin (lim_{x \to 8} x)}{3 - {(lim_{x \to 8} x)}^{2} + \sin (7 {(lim_{x \to 8} x)}^{2})}$

Step 13

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 13.1