The answers to all limits are:
(a) lim x → a [ f(x) + 2g(x)] = -6,
(b) lim x → a [h(x) − 3g(x) + 1] = 13
(c) lim x → a [ f(x)g(x)] = -8
(d) lim x → a [g (x)]² = 16
(e) lim x → a 3√6 + f(x) / 2g(x)
What is the algebraic limit theorem?
The central limit theorem states that if you take sufficiently large samples from a population, the samples' means will be normally distributed, even if the population isn't normally distributed.
here, we have,
(a) lim x → a [ f(x) + 2g(x)]
= lim x → a f(x) + 2 lim x → a g(x) (using algebraic limit theorem)
= 2 + 2(-4)
= -6
(b) lim x → a [h(x) − 3g(x) + 1]
= lim x → a h(x) - 3 lim x → a g(x) + lim x → a 1 (using algebraic limit theorem)
= 0 - 3(-4) + 1
= 13
(c) lim x → a [ f(x)g(x)]
= lim x → a f(x) * lim x → a g(x) (using algebraic limit theorem)
= 2 * (-4)
= -8
(d) lim x → a [g(x)]²
= [lim x → a g(x)]^2 (using algebraic limit theorem)
= (-4)^2
= 16
(e) lim x → a 3√6 + f(x) / 2g(x)
= [3√6 + lim x → a f(x)] / [2 lim x → a g(x)] (using algebraic limit theorem)
= [3√6 + 2] / [2(-4)]
= (-3√6 - 2) / 8
Therefore, the answers to all limits are:
(a) lim x → a [ f(x) + 2g(x)] = -6,
(b) lim x → a [h(x) − 3g(x) + 1] = 13
(c) lim x → a [ f(x)g(x)] = -8
(d) lim x → a [g (x)]² = 16
(e) lim x → a 3√6 + f(x) / 2g(x)
To learn more about algebraic limit theorem visit:
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Complete question:
1. Given that
lim
x → a f(x) = 2, lim
x → a g(x) = −4, lim
x → a h(x) = 0
find the limits.
(a) lim
x → a [ f(x) + 2g(x)]
(b) lim
x → a [h(x) − 3g(x) + 1]
(c) lim
x → a [ f(x)g(x)] (d) lim
x → a [g(x)]2
(e) lim
x → a
3
√6 + f(x) (f ) lim
x → a
2
g(x)
