Calculus Examples

$lim_{x \to 8} \frac{4 e^{x} + 8}{3 e^{x} + 7}$

Step 1

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

$\frac{lim_{x \to 8} 4 e^{x} + 8}{lim_{x \to 8} 3 e^{x} + 7}$

Step 2

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

$\frac{lim_{x \to 8} 4 e^{x} + lim_{x \to 8} 8}{lim_{x \to 8} 3 e^{x} + 7}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 8} e^{x} + lim_{x \to 8} 8}{lim_{x \to 8} 3 e^{x} + 7}$

Step 4

Move the limit into the exponent.

$\frac{4 e^{lim_{x \to 8} x} + lim_{x \to 8} 8}{lim_{x \to 8} 3 e^{x} + 7}$

Step 5

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

$\frac{4 e^{lim_{x \to 8} x} + 8}{lim_{x \to 8} 3 e^{x} + 7}$

Step 6

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

$\frac{4 e^{lim_{x \to 8} x} + 8}{lim_{x \to 8} 3 e^{x} + lim_{x \to 8} 7}$

Step 7

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 e^{lim_{x \to 8} x} + 8}{3 lim_{x \to 8} e^{x} + lim_{x \to 8} 7}$

Step 8

Move the limit into the exponent.

$\frac{4 e^{lim_{x \to 8} x} + 8}{3 e^{lim_{x \to 8} x} + lim_{x \to 8} 7}$

Step 9

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\frac{4 e^{lim_{x \to 8} x} + 8}{3 e^{lim_{x \to 8} x} + 7}$

Step 10

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

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Step 10.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{4 e^{8} + 8}{3 e^{lim_{x \to 8} x} + 7}$

Step 10.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{4 e^{8} + 8}{3 e^{8} + 7}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{4 e^{8} + 8}{3 e^{8} + 7}$

Decimal Form:

$1.33318435 \dots$