Find c so that f(x) is continuouson the entire number line. f(x)={x2if x≤3cxif x&gt;3

Given, f(x)={x2if x≤3cxif x&gt;3 Since the given function is apolynomial function for all x&lt;3 and it is continuous for all x&gt;3 Now we check continuity at x=3 Left hand limit at x=3 is, limx→3−f(x)=limx→3−(x2) =32 =9 Right hand limit at x=3 is, limx→3+f(x)=limx→3+(cx) =c3 For the function to be continuous at x=3, both left hand and right limits must be equal. That is, limx→3−f(x)=limx→3+f(x) ⇒9=c3 ⇒c=27

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BenoguigoliB

Answered question

2021-10-20

Find c so that f(x) is continuouson the entire number line.
$f (x) = {\begin{cases} x^{2} & i f x \leq 3 \\ \frac{c}{x} & i f x > 3 \end{cases}$

Answer & Explanation

Arham Warner

Skilled2021-10-21Added 102 answers

Given,
$f (x) = {\begin{cases} x^{2} & i f x \leq 3 \\ \frac{c}{x} & i f x > 3 \end{cases}$
Since the given function is apolynomial function for all x<3 and it is continuous for all x>3
Now we check continuity at x=3
Left hand limit at x=3 is,
$lim_{x \to 3^{-}} f (x) = lim_{x \to 3^{-}} (x^{2})$
$= 3^{2}$
$= 9$
Right hand limit at x=3 is,
$lim_{x \to 3^{+}} f (x) = lim_{x \to 3^{+}} (\frac{c}{x})$
$= \frac{c}{3}$
For the function to be continuous at x=3, both left hand and right limits must be equal. That is,
$lim_{x \to 3^{-}} f (x) = lim_{x \to 3^{+}} f (x)$
$\Rightarrow 9 = \frac{c}{3}$
$\Rightarrow c = 27$

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