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 $1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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 9

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

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Step 9.1

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 9.2

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 9.3

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

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

Step 10

Simplify the answer.

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Step 10.1

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

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

Step 10.2

Factor $1$ out of $- \tan^{2} (1)$ .

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

Step 10.3

Factor $1$ out of $1 (\sec^{2} (1)) + 1 (- \tan^{2} (1))$ .

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

Step 10.4

Apply pythagorean identity.

$\sec (1 \cdot 1 - 1 \cdot 1)$

Step 10.5

Simplify each term.

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Step 10.5.1

Multiply $1$ by $1$ .

$\sec (1 - 1 \cdot 1)$

Step 10.5.2

Multiply $- 1$ by $1$ .

$\sec (1 - 1)$

Step 10.6

Subtract $1$ from $1$ .

$\sec (0)$

Step 10.7

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

$1$