Calculus Examples

$lim_{x \to 3^{+}} \frac{4 e^{2 x} + 5}{10 - 2 e^{2 x}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3^{+}} 4 e^{2 x} + 5}{lim_{x \to 3^{+}} 10 - 2 e^{2 x}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3^{+}} 4 e^{2 x} + lim_{x \to 3^{+}} 5}{lim_{x \to 3^{+}} 10 - 2 e^{2 x}}$

Step 3

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

$\frac{4 lim_{x \to 3^{+}} e^{2 x} + lim_{x \to 3^{+}} 5}{lim_{x \to 3^{+}} 10 - 2 e^{2 x}}$

Step 4

Move the limit into the exponent.

$\frac{4 e^{lim_{x \to 3^{+}} 2 x} + lim_{x \to 3^{+}} 5}{lim_{x \to 3^{+}} 10 - 2 e^{2 x}}$

Step 5

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

$\frac{4 e^{2 lim_{x \to 3^{+}} x} + lim_{x \to 3^{+}} 5}{lim_{x \to 3^{+}} 10 - 2 e^{2 x}}$

Step 6

Evaluate the limit of $5$ which is constant as $x$ approaches $3$ .

$\frac{4 e^{2 lim_{x \to 3^{+}} x} + 5}{lim_{x \to 3^{+}} 10 - 2 e^{2 x}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{4 e^{2 lim_{x \to 3^{+}} x} + 5}{lim_{x \to 3^{+}} 10 - lim_{x \to 3^{+}} 2 e^{2 x}}$

Step 8

Evaluate the limit of $10$ which is constant as $x$ approaches $3$ .

$\frac{4 e^{2 lim_{x \to 3^{+}} x} + 5}{10 - lim_{x \to 3^{+}} 2 e^{2 x}}$

Step 9

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

$\frac{4 e^{2 lim_{x \to 3^{+}} x} + 5}{10 - 2 lim_{x \to 3^{+}} e^{2 x}}$

Step 10

Move the limit into the exponent.

$\frac{4 e^{2 lim_{x \to 3^{+}} x} + 5}{10 - 2 e^{lim_{x \to 3^{+}} 2 x}}$

Step 11

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

$\frac{4 e^{2 lim_{x \to 3^{+}} x} + 5}{10 - 2 e^{2 lim_{x \to 3^{+}} x}}$

Step 12

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{4 e^{2 \cdot 3} + 5}{10 - 2 e^{2 lim_{x \to 3^{+}} x}}$

Step 12.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{4 e^{2 \cdot 3} + 5}{10 - 2 e^{2 \cdot 3}}$

Step 13

Simplify the answer.

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Step 13.1

Multiply $2$ by $3$ .

$\frac{4 e^{6} + 5}{10 - 2 e^{2 \cdot 3}}$

Step 13.2

Multiply $2$ by $3$ .

$\frac{4 e^{6} + 5}{10 - 2 e^{6}}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{4 e^{6} + 5}{10 - 2 e^{6}}$

Decimal Form:

$- 2.03137323 \dots$