Question

The following limit represents the slope of a curve y=f(x) at the point (a,f(a)). Determine a function f and a number a; then, calculate the limit. √29+h-√29 lim h-0 h GA. Pix) Evh+x OB. f(x)=√h+x-√29 c. f(x)=√x *D. f(x)=√29 Determine the number a. a= (Type an exact answer, using radicals as needed.)

Answer 1

From the limit expression √29+h-√29 lim h-0 h, we can simplify the numerator as:

√(29+h) - √29 = (√(29+h) - √29)(√(29+h) + √29)/(√(29+h) + √29)

= (29+h - 29)/(√(29+h) + √29)

= h/(√(29+h) + √29)

Thus the limit expression becomes:

lim h->0 h/(√(29+h) + √29)

To simplify this expression further, we can multiply the numerator and denominator by the conjugate of the denominator, which is (√(29+h) - √29):

lim h->0 h/(√(29+h) + √29) * (√(29+h) - √29)/(√(29+h) - √29)

= lim h->0 h(√(29+h) - √29)/((29+h) - 29)

= lim h->0 (√(29+h) - √29)/h

This is now in the form of a derivative, specifically the derivative of f(x) = √x evaluated at x = 29. Therefore, we can take f(x) = √x and a = 29, and the limit is the slope of the tangent line to the curve y = √x at x = 29.

To determine the value of the limit, we can use the definition of the derivative:

f'(29) = lim h->0 (f(29+h) - f(29))/h = lim h->0 (√(29+h) - √29)/h

This is the same limit expression we derived earlier. Therefore, f(x) = √x and a = 29, and the limit is f'(29) = lim h->0 (√(29+h) - √29)/h.

To calculate the limit, we can plug in h = 0 and simplify:

lim h->0 (√(29+h) - √29)/h

= lim h->0 ((√(29+h) - √29)/(h))(1/1)

= f'(29)

= 1/(2√29)

Thus, the function f(x) = √x and the number a = 29, and the limit is 1/(2√29).

Answer 2

The function f(x) = √x and number a = 29 satisfy the given limit expression. The limit evaluates to 0.

Step-by-step explanation:

The given limit represents the slope of a curve y=f(x) at the point (a,f(a)). In order to determine the function f and the number a, we can analyze the given options. Option C, f(x) = √x, satisfies the conditions and can be used to calculate the limit.

Substituting the function f(x) = √x into the limit expression, we get:

lim(h→0)(√29+h - √29)/h

Applying the limit laws and simplifying, we can rewrite the expression as:

lim(h→0)√(29+h) - √29/h

Simplifying further, we obtain:

lim(h→0)(√29 - h)/(√(29+h) + √29)

Plugging in the value of h=0, we get:

lim(h→0)√29/(√(29+0) + √29)

Using the sum and difference of squares formula, we can simplify the expression to:

lim(h→0)√29/(√(29+0) + √29) = √29/(√29 + √29)

Finally, we can simplify the expression by rationalizing the denominator:

(√29/(√29 + √29)) × (√29 - √29)/(√29 - √29) = 0