Calculus Examples

$\frac{lim_{n \to 8} x^{2} + x e^{x n} + 1}{e^{n x} + 1}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{n \to 8} x^{2} + lim_{n \to 8} x e^{x n} + lim_{n \to 8} 1}{e^{n x} + 1}$

Step 1.2

Evaluate the limit of $x^{2}$ which is constant as $n$ approaches $8$ .

$\frac{x^{2} + lim_{n \to 8} x e^{x n} + lim_{n \to 8} 1}{e^{n x} + 1}$

Step 1.3

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

$\frac{x^{2} + x lim_{n \to 8} e^{x n} + lim_{n \to 8} 1}{e^{n x} + 1}$

Step 1.4

Move the limit into the exponent.

$\frac{x^{2} + x e^{lim_{n \to 8} x n} + lim_{n \to 8} 1}{e^{n x} + 1}$

Step 1.5

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

$\frac{x^{2} + x e^{x lim_{n \to 8} n} + lim_{n \to 8} 1}{e^{n x} + 1}$

Step 1.6

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

$\frac{x^{2} + x e^{x lim_{n \to 8} n} + 1}{e^{n x} + 1}$

Step 2

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

$\frac{x^{2} + x e^{x \cdot 8} + 1}{e^{n x} + 1}$

Step 3

Move $8$ to the left of $x$ .

$\frac{x^{2} + x e^{8 x} + 1}{e^{n x} + 1}$