Calculus Examples

$lim_{x \to 2} \frac{x + \sqrt{2}}{2}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 2} x + \sqrt{2}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{2} (lim_{x \to 2} x + lim_{x \to 2} \sqrt{2})$

Step 1.3

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

$\frac{1}{2} (lim_{x \to 2} x + \sqrt{2})$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{2} (2 + \sqrt{2})$

Step 3

Simplify the answer.

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Step 3.1

Apply the distributive property.

$\frac{1}{2} \cdot 2 + \frac{1}{2} \sqrt{2}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{1}{2} \cdot 2 + \frac{1}{2} \sqrt{2}$

Step 3.2.2

Rewrite the expression.

$1 + \frac{1}{2} \sqrt{2}$

Step 3.3

Combine $\frac{1}{2}$ and $\sqrt{2}$ .

$1 + \frac{\sqrt{2}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$1 + \frac{\sqrt{2}}{2}$

Decimal Form:

$1.70710678 \dots$