Please, I want to solve all the paragraphs correctly and in clear handwriting.
Limits
A. Compute the following limits:
1. lim(x+1)/(1-x^2)
2. lim(x-3)/(√(x-3))
3. lim(x^2-4x)/(x^2+2x)
B. lim(sin(x))
C. lim(x+sin(x))/(1-cos(x))
Precise statement for limits: Use the definition of the limit of a function to prove the following limits.
a. lim(5x-7) = -2, as x approaches 1.
b. lim(x+1) = 1, as x approaches 0.
Continuity
Check the continuity at the point x = 1, of the function given by:
f(x) = x-2x+1, if 0 < x < 5 and x ≠1
f(x) = 1-x^2, if x = 1
Find real constants a, b, and c so that the following function is continuous on R:
f(x) = (cos(x), 3x+a, bx+3, c), for the function x+1.
Calculate:
a. f(1)
b. The limit as x approaches 1 from the left.
c. The limit as x approaches 1 from the right.
d. The limit as x approaches 1.
From this information, determine if:
i. The function is left continuous.
ii. The function is right continuous at x = 1.
iii. The function f(x) is continuous.
