Calculus Examples

$lim_{y \to 1} \sec (y \sec^{2} (y) - \tan^{2} (y - 1))$

Step 1

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

$\sec (lim_{y \to 1} y \sec^{2} (y) - \tan^{2} (y - 1))$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $1$ .

$\sec (lim_{y \to 1} y \sec^{2} (y) - lim_{y \to 1} \tan^{2} (y - 1))$

Step 3

Split the limit using the Product of Limits Rule on the limit as $y$ approaches $1$ .

$\sec (lim_{y \to 1} y \cdot lim_{y \to 1} \sec^{2} (y) - lim_{y \to 1} \tan^{2} (y - 1))$

Step 4

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

$\sec (lim_{y \to 1} y \cdot {(lim_{y \to 1} \sec (y))}^{2} - lim_{y \to 1} \tan^{2} (y - 1))$

Step 5

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

$\sec (lim_{y \to 1} y \cdot \sec^{2} (lim_{y \to 1} y) - lim_{y \to 1} \tan^{2} (y - 1))$

Step 6

Move the exponent $2$ from $\tan^{2} (y - 1)$ outside the limit using the Limits Power Rule.

$\sec (lim_{y \to 1} y \cdot \sec^{2} (lim_{y \to 1} y) - {(lim_{y \to 1} \tan (y - 1))}^{2})$

Step 7

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

$\sec (lim_{y \to 1} y \cdot \sec^{2} (lim_{y \to 1} y) - \tan^{2} (lim_{y \to 1} y - 1))$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $1$ .

$\sec (lim_{y \to 1} y \cdot \sec^{2} (lim_{y \to 1} y) - \tan^{2} (lim_{y \to 1} y - lim_{y \to 1} 1))$

Step 9

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

$\sec (lim_{y \to 1} y \cdot \sec^{2} (lim_{y \to 1} y) - \tan^{2} (lim_{y \to 1} y - 1 \cdot 1))$

Step 10

Evaluate the limits by plugging in $1$ for all occurrences of $y$ .

Tap for more steps...

Step 10.1

Evaluate the limit of $y$ by plugging in $1$ for $y$ .

$\sec (1 \cdot \sec^{2} (lim_{y \to 1} y) - \tan^{2} (lim_{y \to 1} y - 1 \cdot 1))$

Step 10.2

Evaluate the limit of $y$ by plugging in $1$ for $y$ .

$\sec (1 \cdot \sec^{2} (1) - \tan^{2} (lim_{y \to 1} y - 1 \cdot 1))$

Step 10.3

Evaluate the limit of $y$ by plugging in $1$ for $y$ .

$\sec (1 \cdot \sec^{2} (1) - \tan^{2} (1 - 1 \cdot 1))$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

Multiply $\sec^{2} (1)$ by $1$ .

$\sec (\sec^{2} (1) - \tan^{2} (1 - 1 \cdot 1))$

Step 11.1.2

Evaluate $\sec (1)$ .

$\sec ({1.00015232}^{2} - \tan^{2} (1 - 1 \cdot 1))$

Step 11.1.3

Raise $1.00015232$ to the power of $2$ .

$\sec (1.00030467 - \tan^{2} (1 - 1 \cdot 1))$

Step 11.1.4

Multiply $- 1$ by $1$ .

$\sec (1.00030467 - \tan^{2} (1 - 1))$

Step 11.1.5

Subtract $1$ from $1$ .

$\sec (1.00030467 - \tan^{2} (0))$

Step 11.1.6

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

$\sec (1.00030467 - 0^{2})$

Step 11.1.7

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

$\sec (1.00030467 - 0)$

Step 11.1.8

Multiply $- 1$ by $0$ .

$\sec (1.00030467 + 0)$

Step 11.2

Add $1.00030467$ and $0$ .

$\sec (1.00030467)$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\sec (1.00030467)$

Decimal Form:

$1.85169445 \dots$