If lim x - a fx-L and lim x - a gx-m- then lim x - afxgx-LmA- FalseB- True

The correct option is A False If both the limits lim_x→ af(x) and lim_x→ ag(x) exists finitely, then we can apply the limits independently to them in a limt i ...

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Byju's Answer

Standard XII

Mathematics

Derivative from First Principle

If lim x → a ...

Question

If ${lim}_{x \to a} f (x) = L a n d {lim}_{x \to a} g (x) = m$ , then ${lim}_{x \to a} (f (x))^{g (x)} = L^{m}$

True

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False

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Solution

The correct option is B
False

If both the limits ${lim}_{x \to a} f (x) a n d {lim}_{x \to a} g (x)$ exists finitely, then we can apply the limits independently to them in a limt involving both of them and $x \to a$ , provided the limit does not become indeterminate or infinite.
This is true for taking powers also, like $f (x)^{g (x)}$ . We do not need the exact proof of this. A fair intuition is good enough to understand coming concepts and solve problems.

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Similar questions

If $lim x \to a f (x) = l$ and $lim x \to a g (x) = m$ , then $lim x \to a {(f (x))}^{g (x)} = l^{m}$

If both $lim x \to a f (x)$ and $lim x \to a g (x)$ and exist finitely and $lim x \to a g (x) = 0$ , then
$lim x \to a \frac{f (x)}{g (x)} = \frac{lim x \to a f (x)}{lim x \to a g (x)}$

If $lim x \to a [f (x) + g (x)] = 10$ and $lim x \to a f (x) = 2$ , then find the value of $lim x \to a g (x)$ , provided that $lim x \to a f (x)$ and $lim x \to a g (x)$ exists ___

Q. Assertion :If

a_{i} > 0 \forall (i = 1, 2, 3, . . ., n)

then

lim x \to \infty {⎧ ⎨ ⎩ \frac{\sum_{i = 1}^{n} {(a_{i})}_{i}^{1 / x}}{n} ⎫ ⎬ ⎭}^{n x} = n \prod i = 1 a_{i}

Reason: If

lim x \to a f (x) = 1

lim x \to a g (x) = \infty

, then

lim x \to a {f (x)}^{g (x)} = e^{lim x \to a {f (x) - 1}} \times g (x)

If both $lim x \to a f (x)$ and $lim x \to a g (x)$ and exist finitely and $lim x \to a g (x) = 0$ , then
$lim x \to a \frac{f (x)}{g (x)} = \frac{lim x \to a f (x)}{lim x \to a g (x)}$

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If limx→af(x)=L and limx→ag(x)=m, then limx→a(f(x))g(x)=Lm

If ${lim}_{x \to a} f (x) = L a n d {lim}_{x \to a} g (x) = m$ , then ${lim}_{x \to a} (f (x))^{g (x)} = L^{m}$