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How do you find the limit of #(sqrt(1+h)-1)/h # as h approaches 0?

Answer 1

I found: #1/2#

If you try the limit directly you'll get the indeterminate form #0/0#;
To avoid this I 'd try a little manipulation, a kind of "strange" inverse rationalization (orange circle). Have a look:

Graphically
graph{(sqrt(1+x)-1)/x [-10, 10, -5, 5]}

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Answer 2

Samantha Adkison

To find the limit of (sqrt(1+h)-1)/h as h approaches 0, we can use the concept of limits and algebraic manipulation.

First, let's simplify the expression by rationalizing the numerator.

Multiply both the numerator and denominator by the conjugate of the numerator, which is sqrt(1+h) + 1.

This gives us ((sqrt(1+h)-1)/h) * ((sqrt(1+h)+1)/(sqrt(1+h)+1)).

Simplifying further, we have (1+h-1)/(h(sqrt(1+h)+1)).

The numerator simplifies to h, and the denominator becomes h(sqrt(1+h)+1).

Now, we can cancel out the h terms in the numerator and denominator.

This leaves us with 1/(sqrt(1+h)+1).

As h approaches 0, the expression becomes 1/(sqrt(1+0)+1), which simplifies to 1/2.

Therefore, the limit of (sqrt(1+h)-1)/h as h approaches 0 is 1/2.

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Answer from HIX Tutor

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.