Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Simplify the answer.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Apply pythagorean identity.

$1 \cdot 1 - 1 \cdot 1$

Step 9.5

Simplify each term.

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Step 9.5.1

Multiply $1$ by $1$ .

$1 - 1 \cdot 1$

Step 9.5.2

Multiply $- 1$ by $1$ .

$1 - 1$

Step 9.6

Subtract $1$ from $1$ .

$0$