Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (x^{2})$ in terms of sines and cosines.

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

Step 1.2

Rewrite $\sec (x - 1)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{1}{\cos (x - 1)}}{\frac{\sin (x^{2})}{\cos (x^{2})}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x^{2})}{\cos (x^{2})}$ .

$lim_{x \to 0} \frac{1}{\cos (x - 1)} \cdot \frac{\cos (x^{2})}{\sin (x^{2})}$

Step 1.4

Simplify.

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Step 1.4.1

Convert from $\frac{1}{\cos (x - 1)}$ to $\sec (x - 1)$ .

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

Step 1.4.2

Convert from $\frac{\cos (x^{2})}{\sin (x^{2})}$ to $\cot (x^{2})$ .

$lim_{x \to 0} \sec (x - 1) \cot (x^{2})$

Step 2

Evaluate the limit.

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Step 2.1

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

$lim_{x \to 0} \sec (x - 1) \cdot lim_{x \to 0} \cot (x^{2})$

Step 2.2

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

$\sec (lim_{x \to 0} x - 1) \cdot lim_{x \to 0} \cot (x^{2})$

Step 2.3

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

$\sec (lim_{x \to 0} x - lim_{x \to 0} 1) \cdot lim_{x \to 0} \cot (x^{2})$

Step 2.4

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

$\sec (lim_{x \to 0} x - 1 \cdot 1) \cdot lim_{x \to 0} \cot (x^{2})$

Step 3

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

$\sec (0 - 1 \cdot 1) \cdot lim_{x \to 0} \cot (x^{2})$

Step 4

Consider the left sided limit.

$lim_{x \to 0^{-}} \cot (x^{2})$

Step 5

As the $x$ values approach $0$ from the left, the function values increase without bound.

$\infty$

Step 6

Consider the right sided limit.

$lim_{x \to 0^{+}} \cot (x^{2})$

Step 7

As the $x$ values approach $0$ from the right, the function values increase without bound.

$\sec (0 - 1 \cdot 1) \cdot \infty$

Step 8

Simplify the answer.

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Step 8.1

Multiply $- 1$ by $1$ .

$\sec (0 - 1) \cdot \infty$

Step 8.2

Subtract $1$ from $0$ .

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

Step 8.3

Evaluate $\sec (- 1)$ .

$1.00015232 \cdot \infty$

Step 8.4

A non-zero constant times infinity is infinity.

$\infty$