Calculus Examples

$lim_{x \to \frac{π}{2}} 2 x + \sin (x)$

Step 1

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

$lim_{x \to \frac{π}{2}} 2 x + lim_{x \to \frac{π}{2}} \sin (x)$

Step 2

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

$2 lim_{x \to \frac{π}{2}} x + lim_{x \to \frac{π}{2}} \sin (x)$

Step 3

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

$2 lim_{x \to \frac{π}{2}} x + \sin (lim_{x \to \frac{π}{2}} x)$

Step 4

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

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Step 4.1

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

$2 \frac{π}{2} + \sin (lim_{x \to \frac{π}{2}} x)$

Step 4.2

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

$2 \frac{π}{2} + \sin (\frac{π}{2})$

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $2$ .

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Step 5.1.1

Cancel the common factor.

$2 \frac{π}{2} + \sin (\frac{π}{2})$

Step 5.1.2

Rewrite the expression.

$π + \sin (\frac{π}{2})$

Step 5.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$π + 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$π + 1$

Decimal Form:

$4.14159265 \dots$