Calculus Examples

$lim_{x \to \frac{π}{4}} \frac{1}{\tan (x - 1)} - \frac{2}{\tan^{2} (x) - 1}$

Step 1

Apply trigonometric identities.

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Step 1.1

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

$lim_{x \to \frac{π}{4}} \frac{1}{\frac{\sin (x - 1)}{\cos (x - 1)}} - \frac{2}{\tan^{2} (x) - 1}$

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x - 1)}{\cos (x - 1)}$ .

$lim_{x \to \frac{π}{4}} \frac{\cos (x - 1)}{\sin (x - 1)} - \frac{2}{\tan^{2} (x) - 1}$

Step 1.3

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

$lim_{x \to \frac{π}{4}} \cot (x - 1) - \frac{2}{\tan^{2} (x) - 1}$

Step 2

Combine terms.

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Step 2.1

To write $\cot (x - 1)$ as a fraction with a common denominator, multiply by $\frac{\tan^{2} (x) - 1}{\tan^{2} (x) - 1}$ .

$lim_{x \to \frac{π}{4}} \frac{\cot (x - 1) (\tan^{2} (x) - 1)}{\tan^{2} (x) - 1} - \frac{2}{\tan^{2} (x) - 1}$

Step 2.2

Combine the numerators over the common denominator.

$lim_{x \to \frac{π}{4}} \frac{\cot (x - 1) (\tan^{2} (x) - 1) - 2}{\tan^{2} (x) - 1}$

Step 3

Since the function approaches $\infty$ from the left and $- \infty$ from the right, the limit does not exist.

$Does not exist$