Evaluate the limit as x approaches 1 of the quotient of the square root of the quantity x squared plus 3 minus 2 and the quantity x minus 1. You must show your work to receive credit?

1 Answer
Anjali G
Feb 4, 2018

#lim_(x->1) frac{sqrt(x^2 + 3) - 2}{x-1} = 1/2#

Explanation:

Let me know if I interpreted the problem incorrectly. I interpreted it as:
#lim_(x->1) frac{sqrt(x^2 + 3) - 2}{x-1}#

By evaluating directly, we get #0/0#. So, we apply L'Hospital's rule:
This limit is the same as the limit of the derivative of the numerator over the derivative of the denominator as x approaches 1.

# =^"H" lim_(x->1) frac{d/dx(sqrt(x^2 + 3) - 2)}{d/dx (x-1)}#

# = lim_(x->1) frac{1/2(x^2+3)^(-1/2)(2x)}{1}#

# = lim_(x->1) frac{x}{sqrt(x^2+3)}#

# =1/(sqrt(1^2+3))#

# = 1/2#

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