Calculus Examples

$lim_{x \to 8} \frac{x^{101} + 9^{x} - e^{\sqrt{x}}}{2 \cdot 9^{x} - 10^{6} \sin (x) + x^{25}}$

Step 1

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

$\frac{lim_{x \to 8} x^{101} + 9^{x} - e^{\sqrt{x}}}{lim_{x \to 8} 2 \cdot 9^{x} - 10^{6} \sin (x) + x^{25}}$

Step 2

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

$\frac{lim_{x \to 8} x^{101} + lim_{x \to 8} 9^{x} - lim_{x \to 8} e^{\sqrt{x}}}{lim_{x \to 8} 2 \cdot 9^{x} - 10^{6} \sin (x) + x^{25}}$

Step 3

Move the exponent $101$ from $x^{101}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 8} x)}^{101} + lim_{x \to 8} 9^{x} - lim_{x \to 8} e^{\sqrt{x}}}{lim_{x \to 8} 2 \cdot 9^{x} - 10^{6} \sin (x) + x^{25}}$

Step 4

Move the limit into the exponent.

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - lim_{x \to 8} e^{\sqrt{x}}}{lim_{x \to 8} 2 \cdot 9^{x} - 10^{6} \sin (x) + x^{25}}$

Step 5

Move the limit into the exponent.

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{lim_{x \to 8} \sqrt{x}}}{lim_{x \to 8} 2 \cdot 9^{x} - 10^{6} \sin (x) + x^{25}}$

Step 6

Move the limit under the radical sign.

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{\sqrt{lim_{x \to 8} x}}}{lim_{x \to 8} 2 \cdot 9^{x} - 10^{6} \sin (x) + x^{25}}$

Step 7

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

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{\sqrt{lim_{x \to 8} x}}}{lim_{x \to 8} 2 \cdot 9^{x} - lim_{x \to 8} 10^{6} \sin (x) + lim_{x \to 8} x^{25}}$

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{\sqrt{lim_{x \to 8} x}}}{2 lim_{x \to 8} 9^{x} - lim_{x \to 8} 10^{6} \sin (x) + lim_{x \to 8} x^{25}}$

Step 9

Move the limit into the exponent.

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{\sqrt{lim_{x \to 8} x}}}{2 \cdot 9^{lim_{x \to 8} x} - lim_{x \to 8} 10^{6} \sin (x) + lim_{x \to 8} x^{25}}$

Step 10

Move the term $10^{6}$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{\sqrt{lim_{x \to 8} x}}}{2 \cdot 9^{lim_{x \to 8} x} - 10^{6} lim_{x \to 8} \sin (x) + lim_{x \to 8} x^{25}}$

Step 11

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

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{\sqrt{lim_{x \to 8} x}}}{2 \cdot 9^{lim_{x \to 8} x} - 10^{6} \sin (lim_{x \to 8} x) + lim_{x \to 8} x^{25}}$

Step 12

Move the exponent $25$ from $x^{25}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 8} x)}^{101} + 9^{lim_{x \to 8} x} - e^{\sqrt{lim_{x \to 8} x}}}{2 \cdot 9^{lim_{x \to 8} x} - 10^{6} \sin (lim_{x \to 8} x) + {(lim_{x \to 8} x)}^{25}}$

Step 13

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

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Step 13.1