Calculus Examples

$lim_{x \to 0} \frac{7 + 2 x}{\cos (x)}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 7 + 2 x}{lim_{x \to 0} \cos (x)}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 7 + lim_{x \to 0} 2 x}{lim_{x \to 0} \cos (x)}$

Step 3

Evaluate the limit of $7$ which is constant as $x$ approaches $0$ .

$\frac{7 + lim_{x \to 0} 2 x}{lim_{x \to 0} \cos (x)}$

Step 4

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

$\frac{7 + 2 lim_{x \to 0} x}{lim_{x \to 0} \cos (x)}$

Step 5

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

$\frac{7 + 2 lim_{x \to 0} x}{\cos (lim_{x \to 0} x)}$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{7 + 2 \cdot 0}{\cos (lim_{x \to 0} x)}$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{7 + 2 \cdot 0}{\cos (0)}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $2$ by $0$ .

$\frac{7 + 0}{\cos (0)}$

Step 7.1.2

Add $7$ and $0$ .

$\frac{7}{\cos (0)}$

Step 7.2

The exact value of $\cos (0)$ is $1$ .

$\frac{7}{1}$

Step 7.3

Divide $7$ by $1$ .

$7$