Find the values of a and b such that the function f defined by

is a continuous function at x = 4.

Question

Find the values of a and b such that the function f defined by is a continuous function at x = 4.

Philoid · Accepted Answer

Given,

…(1)

We need to find the value of a & b such that f(x) is continuous at x = 4.

A function f(x) is said to be continuous at x = c if,

Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).

Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)

Now, let’s assume that f(x) is continuous at x = 4.

∴

As we have to find a & b, so pick out a combination so that we get a or b in our equation.

In this question first we take LHL = f(4)

∴

⇒ {using equation 1}

⇒

∵ h > 0 as defined in theory above.

∴ |-h| = h

⇒

⇒ a – 1 = a + b

∴ b = -1

Now, taking other combination,

RHL = f(4)

⇒ {using equation 1}

⇒

∵ h > 0 as defined in theory above.

∴ |h| = h

⇒

⇒ b + 1 = a + b

∴ a = 1

Hence,

a = 1 and b = -1

Find the values of a and b such that the function f defined byis a continuous function at x = 4.