Calculus Examples

$\frac{lim_{x \to 0} 1 - \sec^{2} (7 x)}{{(3 x)}^{2}}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \sec^{2} (7 x)}{{(3 x)}^{2}}$

Step 1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1 - lim_{x \to 0} \sec^{2} (7 x)}{{(3 x)}^{2}}$

Step 1.3

Move the exponent $2$ from $\sec^{2} (7 x)$ outside the limit using the Limits Power Rule.

$\frac{1 - {(lim_{x \to 0} \sec (7 x))}^{2}}{{(3 x)}^{2}}$

Step 1.4

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

$\frac{1 - \sec^{2} (lim_{x \to 0} 7 x)}{{(3 x)}^{2}}$

Step 1.5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 - \sec^{2} (7 lim_{x \to 0} x)}{{(3 x)}^{2}}$

Step 2

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

$\frac{1 - \sec^{2} (7 \cdot 0)}{{(3 x)}^{2}}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Reorder $1$ and $- \sec^{2} (7 \cdot 0)$ .

$\frac{- \sec^{2} (7 \cdot 0) + 1}{{(3 x)}^{2}}$

Step 3.2

Factor $- 1$ out of $- \sec^{2} (7 \cdot 0)$ .

$\frac{- (\sec^{2} (7 \cdot 0)) + 1}{{(3 x)}^{2}}$

Step 3.3

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- \sec^{2} (7 \cdot 0) - 1 \cdot - 1}{{(3 x)}^{2}}$

Step 3.4

Factor $- 1$ out of $- \sec^{2} (7 \cdot 0) - 1 \cdot - 1$ .

$\frac{- (\sec^{2} (7 \cdot 0) - 1)}{{(3 x)}^{2}}$

Step 3.5

Apply pythagorean identity.

$\frac{- \tan^{2} (7 \cdot 0)}{{(3 x)}^{2}}$

Step 3.6

Simplify the numerator.

Tap for more steps...

Step 3.6.1

Multiply $7$ by $0$ .

$\frac{- \tan^{2} (0)}{{(3 x)}^{2}}$

Step 3.6.2

The exact value of $\tan (0)$ is $0$ .

$\frac{- 0^{2}}{{(3 x)}^{2}}$

Step 3.6.3

Raising $0$ to any positive power yields $0$ .

$\frac{- 0}{{(3 x)}^{2}}$

Step 3.7

Simplify the denominator.

Tap for more steps...

Step 3.7.1

Apply the product rule to $3 x$ .

$\frac{- 0}{3^{2} x^{2}}$

Step 3.7.2

Raise $3$ to the power of $2$ .

$\frac{- 0}{9 x^{2}}$

Step 3.8

Multiply $- 1$ by $0$ .

$\frac{0}{9 x^{2}}$

Step 3.9

Cancel the common factor of $0$ and $9$ .

Tap for more steps...

Step 3.9.1

Factor $9$ out of $0$ .

$\frac{9 (0)}{9 x^{2}}$

Step 3.9.2

Cancel the common factors.

Tap for more steps...

Step 3.9.2.1

Factor $9$ out of $9 x^{2}$ .

$\frac{9 (0)}{9 (x^{2})}$

Step 3.9.2.2

Cancel the common factor.

$\frac{9 \cdot 0}{9 x^{2}}$

Step 3.9.2.3

Rewrite the expression.

$\frac{0}{x^{2}}$

Step 3.10

Divide $0$ by $x^{2}$ .

$0$