Find #a# and #b# so that # f(x) = { (x^2, xle1),(ax+b,x gt 1) :} # is continuous?

1 Answer
Steve M
Mar 27, 2017

# a=1-b #

Explanation:

We have:

# f(x) = { (x^2, xle1),(ax+b,x gt 1) :} #

Both #x^2# and #ax+b# are polynomials, and so are continuous in there own right, so the only possibility of a discontinuity is at the junction between the two polynomials at #x=1#

In order for continuity at #x=1# then we require (by the definition of continuity):

# lim_(x rarr 1) f(x) = f(1) #

Consider the LH limit;

# lim_(x rarr 1^-) f(x) = lim_(x rarr 1^-) x^2 = 1#

And the RH limit:

# lim_(x rarr 1^+) f(x) = lim_(x rarr 1^+) ax+b =a+b#

So the requirement for continuity is:

# lim_(x rarr 1^-) f(x) = lim_(x rarr 1^+) f(x) #

# :. a+b =1 => a=1-b #

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