# Evaluate the limit #lim_(x->0) (e^x+3x)^(1/x) #?

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Write the function as:

Consider now the limit:

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The limit of (e^x+3x)^(1/x) as x approaches 0 is e^3.

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To evaluate the limit lim_(x->0) (e^x+3x)^(1/x), we can rewrite the expression using properties of logarithms and exponential functions:

lim_(x->0) (e^x+3x)^(1/x) = lim_(x->0) e^(1/x * ln(e^x + 3x))

Now, we can use the fact that ln(a^b) = b * ln(a):

= lim_(x->0) e^(ln(e^x + 3x)/x)

Applying L'Hôpital's Rule, where we differentiate the numerator and denominator separately:

= lim_(x->0) (d/dx(ln(e^x + 3x)))/(d/dx(x))

= lim_(x->0) ((e^x + 3)/(e^x + 3x))/(1)

Now, substitute x = 0 into the expression:

= ((e^0 + 3)/(e^0 + 3*0))/(1) = (1 + 3)/(1) = 4

So, the limit lim_(x->0) (e^x+3x)^(1/x) is equal to 4.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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