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Find 
lim_(x rarr0)g(x) for

g(x)=(4x^(2)-15)/(4x+3)". "

Find limx0g(x) \lim _{x \rightarrow 0} g(x) for\newlineg(x)=4x2154x+3 g(x)=\frac{4 x^{2}-15}{4 x+3} \text {. }

Full solution

Q. Find limx0g(x) \lim _{x \rightarrow 0} g(x) for\newlineg(x)=4x2154x+3 g(x)=\frac{4 x^{2}-15}{4 x+3} \text {. }
  1. Identify Function and Point: Identify the function and the point at which we need to find the limit.\newlineWe are given the function g(x)=4x2154x+3g(x) = \frac{4x^2 - 15}{4x + 3} and we need to find the limit as xx approaches 00.
  2. Direct Substitution: Direct substitution to check if the limit can be found easily.\newlineLet's substitute x=0x = 0 into the function g(x)g(x) to see if we get a determinate form.\newlineg(0)=4(0)2154(0)+3=153=5g(0) = \frac{4(0)^2 - 15}{4(0) + 3} = \frac{-15}{3} = -5\newlineSince we get a real number and not an indeterminate form, the limit exists and is equal to 5-5.
  3. Conclude Limit: Conclude the limit based on the substitution.\newlineSince the direct substitution of x=0x = 0 into g(x)g(x) gave us a real number, we can conclude that the limit of g(x)g(x) as xx approaches 00 is 5-5.

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