calculate the lim as x-> pi (cos(x)+1)/(x-pi) using the special limit
lim x->0 sinx/x
To calculate the limit as x approaches pi of (cos(x)+1)/(x-pi) using the special limit lim x->0 sinx/x, we can follow these steps:
Step 1: Rewrite the original expression using the special limit.
lim x->pi (cos(x)+1)/(x-pi)
= lim x->pi (cos(x)+1) / [sin(x)/(x-pi)]
Step 2: Simplify the expression by multiplying the numerator and denominator by (x-pi).
lim x->pi (cos(x)+1)(x-pi) / sin(x)
Step 3: Expand the numerator and distribute.
lim x->pi (cos(x)(x-pi) + (x-pi)) / sin(x)
Step 4: Expand further and rearrange the terms.
lim x->pi (cos(x)x - cos(x)pi + x - pi) / sin(x)
Step 5: Factor out the common term (x - pi) in the numerator.
lim x->pi [(cos(x) - 1)(x - pi) + (x - pi)] / sin(x)
Step 6: Cancel out the common factor (x - pi) in the numerator and denominator.
lim x->pi (cos(x) - 1 + 1) / sin(x)
= lim x->pi cos(x) / sin(x)
Step 7: Use the special limit lim x->0 sinx/x to evaluate the new expression.
= cos(pi) / sin(pi)
= -1 / 0
The limit of (cos(x)+1)/(x-pi) as x approaches pi is undefined because the denominator approaches zero (0) but the numerator does not.