Calculus Examples

$lim_{x \to \frac{1}{2}} \frac{x \cos (π x)}{e^{x} - \sqrt{e}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x \cos (π x)}{lim_{x \to \frac{1}{2}} e^{x} - \sqrt{e}}$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x \cdot lim_{x \to \frac{1}{2}} \cos (π x)}{lim_{x \to \frac{1}{2}} e^{x} - \sqrt{e}}$

Step 3

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

$\frac{lim_{x \to \frac{1}{2}} x \cdot \cos (lim_{x \to \frac{1}{2}} π x)}{lim_{x \to \frac{1}{2}} e^{x} - \sqrt{e}}$

Step 4

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

$\frac{lim_{x \to \frac{1}{2}} x \cdot \cos (π lim_{x \to \frac{1}{2}} x)}{lim_{x \to \frac{1}{2}} e^{x} - \sqrt{e}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x \cdot \cos (π lim_{x \to \frac{1}{2}} x)}{lim_{x \to \frac{1}{2}} e^{x} - lim_{x \to \frac{1}{2}} \sqrt{e}}$

Step 6

Move the limit into the exponent.

$\frac{lim_{x \to \frac{1}{2}} x \cdot \cos (π lim_{x \to \frac{1}{2}} x)}{e^{lim_{x \to \frac{1}{2}} x} - lim_{x \to \frac{1}{2}} \sqrt{e}}$

Step 7

Evaluate the limit of $\sqrt{e}$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{lim_{x \to \frac{1}{2}} x \cdot \cos (π lim_{x \to \frac{1}{2}} x)}{e^{lim_{x \to \frac{1}{2}} x} - \sqrt{e}}$

Step 8

Evaluate the limits by plugging in $\frac{1}{2}$ for all occurrences of $x$ .

Tap for more steps...

Step 8.1

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{\frac{1}{2} \cdot \cos (π lim_{x \to \frac{1}{2}} x)}{e^{lim_{x \to \frac{1}{2}} x} - \sqrt{e}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{\frac{1}{2} \cdot \cos (π \frac{1}{2})}{e^{lim_{x \to \frac{1}{2}} x} - \sqrt{e}}$

Step 8.3

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{\frac{1}{2} \cdot \cos (π \frac{1}{2})}{e^{\frac{1}{2}} - \sqrt{e}}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Combine $\frac{1}{2}$ and $\cos (π \frac{1}{2})$ .

$\frac{\frac{\cos (π \frac{1}{2})}{2}}{e^{\frac{1}{2}} - \sqrt{e}}$

Step 9.2

Simplify the numerator.

Tap for more steps...

Step 9.2.1

Combine $π$ and $\frac{1}{2}$ .

$\frac{\frac{\cos (\frac{π}{2})}{2}}{e^{\frac{1}{2}} - \sqrt{e}}$

Step 9.2.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{\frac{0}{2}}{e^{\frac{1}{2}} - \sqrt{e}}$

Step 9.3

Divide $0$ by $2$ .

$\frac{0}{e^{\frac{1}{2}} - \sqrt{e}}$

Step 9.4

Divide $0$ by $e^{\frac{1}{2}} - \sqrt{e}$ .

$0$