Calculus Examples

$lim_{x \to 8} \frac{5 + e^{4 x}}{1 + \ln (x)}$

Step 1

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

$\frac{lim_{x \to 8} 5 + e^{4 x}}{lim_{x \to 8} 1 + \ln (x)}$

Step 2

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

$\frac{lim_{x \to 8} 5 + lim_{x \to 8} e^{4 x}}{lim_{x \to 8} 1 + \ln (x)}$

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{5 + lim_{x \to 8} e^{4 x}}{lim_{x \to 8} 1 + \ln (x)}$

Step 4

Move the limit into the exponent.

$\frac{5 + e^{lim_{x \to 8} 4 x}}{lim_{x \to 8} 1 + \ln (x)}$

Step 5

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

$\frac{5 + e^{4 lim_{x \to 8} x}}{lim_{x \to 8} 1 + \ln (x)}$

Step 6

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

$\frac{5 + e^{4 lim_{x \to 8} x}}{lim_{x \to 8} 1 + lim_{x \to 8} \ln (x)}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{5 + e^{4 lim_{x \to 8} x}}{1 + lim_{x \to 8} \ln (x)}$

Step 8

Move the limit inside the logarithm.

$\frac{5 + e^{4 lim_{x \to 8} x}}{1 + \ln (lim_{x \to 8} x)}$

Step 9

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

Tap for more steps...

Step 9.1

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

$\frac{5 + e^{4 \cdot 8}}{1 + \ln (lim_{x \to 8} x)}$

Step 9.2

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

$\frac{5 + e^{4 \cdot 8}}{1 + \ln (8)}$

Step 10

Multiply $4$ by $8$ .

$\frac{5 + e^{32}}{1 + \ln (8)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{5 + e^{32}}{1 + \ln (8)}$

Decimal Form:

$25641974076770.8 \dots$