Calculus Examples

$lim_{x \to 0.5} \cos (x) + 2 \sin (3 x)$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0.5$ .

$lim_{x \to 0.5} \cos (x) + lim_{x \to 0.5} 2 \sin (3 x)$

Step 2

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

$\cos (lim_{x \to 0.5} x) + lim_{x \to 0.5} 2 \sin (3 x)$

Step 3

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

$\cos (lim_{x \to 0.5} x) + 2 lim_{x \to 0.5} \sin (3 x)$

Step 4

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

$\cos (lim_{x \to 0.5} x) + 2 \sin (lim_{x \to 0.5} 3 x)$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\cos (lim_{x \to 0.5} x) + 2 \sin (3 lim_{x \to 0.5} x)$

Step 6

Evaluate the limits by plugging in $0.5$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $0.5$ for $x$ .

$\cos (0.5) + 2 \sin (3 lim_{x \to 0.5} x)$

Step 6.2

Evaluate the limit of $x$ by plugging in $0.5$ for $x$ .

$\cos (0.5) + 2 \sin (3 \cdot 0.5)$

Step 7

Multiply $3$ by $0.5$ .

$\cos (0.5) + 2 \sin (1.5)$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\cos (0.5) + 2 \sin (1.5)$

Decimal Form:

$2.87257253 \dots$