find a and b such that
lim (square root ax + b) - 2/ x = 3
x->0
To find the values of a and b such that the given limit is 3 as x approaches 0, we can use limit properties and algebraic manipulation. Here's how you can solve it step by step:
Step 1: Start by rewriting the given limit expression with a square root as a fraction:
lim(x->0) [(√(ax + b) - 2)/x] = 3
Step 2: Multiply both sides of the equation by x:
lim(x->0) (√(ax + b) - 2) = 3x
Step 3: Square both sides of the equation to eliminate the square root:
[lim(x->0) (√(ax + b) - 2)]^2 = (3x)^2
[lim(x->0) (√(ax + b) - 2)]^2 = 9x^2
Step 4: Expand the left side of the equation:
lim(x->0) (ax + b - 4√(ax + b) + 4) = 9x^2
Step 5: Take the limit as x approaches 0 of each term separately:
[lim(x->0) (ax)] + [lim(x->0) (b)] - [lim(x->0) (4√(ax + b))] + [lim(x->0) (4)] = 9x^2
Step 6: Simplify the terms:
0 + b - 4√(0 + b) + 4 = 0
b - 4√b + 4 = 0
Step 7: Solve the equation for b:
b + 4 - 4√b = 0
4 - 4√b = -b
4 + b = 4√b
(4 + b)^2 = (4√b)^2
Step 8: Expand and simplify the equation:
16 + 8b + b^2 = 16b
b^2 + 8b + 16 = 16b
Step 9: Rearrange the equation and solve for b:
b^2 - 8b + 16 = 0
(b - 4)^2 = 0
b - 4 = 0
b = 4
Step 10: Substitute the value of b back into the original equation to find a:
lim(x->0) (√(ax + 4) - 2)/x = 3
lim(x->0) (√(ax + 4) - 2) = 3x
√(a(0) + 4) - 2 = 0
√4 - 2 = 0
2 - 2 = 0
0 = 0
Since we get 0 = 0 when substituting x = 0 into the equation, we can conclude that any value of a would satisfy the equation.
Therefore, a can be any real number, and b is equal to 4.