Find values of x, if any, at which f is not continuous: f(x)=5x+2xx+4

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l1koV · Accepted Answer

A function is continuous if the left-hand limit at x=c and the right-hand limit at x=c both are equal to the value of the function at x=c. limx→c−f(x)=limx→c+f(x)=f(c) In other words, a function is continuous if its graph is a single and non-broken curve or line. If a function has a vertical asymptote at x=b, then the function is not defined for x=b. It means the function is discontinuous at x=b. To find the vertical asymptotes, we need to find the values of x for which the denominator value is 0. The given function is: f(x)=5x+2xx+4 Taking LCM, we get f(x)=5(x+4)+2x(x)x(x+4) f(x)=5x+20+2x2x(x+4) Here, the numerator and denominator both are polynomials, and the polynomials are always continuous. So, the given function of continuous for all values of x except the values for which the denominator is equal to 0. x(x+4)=0 x=0 and x+4=0 x=0 and x=−4 It means the given function is not defined for x=0 and x=−4. So, the vertical asymptotes of the given function are x=0 and x=−4. The function is discontinuous at the vertical asymptotes. So, the given function is discontinuous at x=0 and x=−4. Answer: The function f(x) is discontinuous or not continuous at x=0 and x=−4.

Find values of x, if any, at which f is not continuous: f(x)=\frac{5}{x}+\frac{2x}{x+4}

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