How would you find the limit of (secx-1)/(x^2) as x goes to 0 algebraically?
change secx -1 to 1/cosx -1 to (1-cosx)/cosx
then
lim (1-cosx)/(cosx)x^2 rationalize the numberator.
lim (1-cosx)(1+cosx)/(1+cosx)cosx x^2
lim (sin^2x)/x^2 * 1/(1+cosx)cosx
lim (sinx/x)lim sinx/x Lim (1/(cosx)(1+cosx)
1*1*1/(1*2)= 1/2
Multiply top and bottom by (sec(x)+1) to get
(secx-1)(sec(x)+1)/((x^2)(sec(x)+1)
=(sec²(x)-1)/((x^2)(sec(x)+1)
=(sin(x)/x)²/(cos²(x)(1+sec(x)))
Lim x->0 (sin(x)/x)=1
Lim x->0 cos(x)=1
Lim x->0 sec(x)=1
Therefore
Lim x->0 (secx-1)/(x^2) = 1/2
Alternatively, use l'Hôpital's rule.
To find the limit of (secx-1)/(x^2) as x approaches 0 algebraically, you can use L'Hôpital's rule. This rule states that if you have a limit of the form 0/0 or ∞/∞, you can take the derivative of the numerator and denominator separately and then evaluate the limit again.
Let's apply L'Hôpital's rule to find the limit:
1. Take the derivative of the numerator and denominator separately:
The derivative of secx is secx * tanx, and the derivative of x^2 is 2x.
So, the expression becomes (secx * tanx)/(2x).
2. Simplify the expression:
Distribute secx and cancel out an x from the numerator and denominator:
(secx * tanx)/(2x) = (tanx)/(2) = (sinx/cosx)/(2) = sinx/(2cosx).
3. Evaluate the limit as x approaches 0:
Plugging in x = 0, we get sin0/(2cos0) = 0/2 = 0.
Therefore, the limit of (secx-1)/(x^2) as x approaches 0 algebraically is 0.
To find the limit of (secx-1)/(x^2) as x approaches 0 algebraically, we can use a few algebraic manipulation techniques:
Step 1: Recall the identity secx = 1/cosx.
Step 2: Substitute 1/cosx for secx in the expression. Our expression now becomes (1/cosx - 1)/(x^2).
Step 3: Combine the fractions by getting a common denominator. Our expression can be rewritten as (1 - cosx)/(x^2 * cosx).
Step 4: Using the identity cosx = 1 - (1/2)x^2 + (1/24)x^4 - ..., we can rewrite cosx as 1 - (1/2)x^2 + higher order terms.
Step 5: Substituting the expression for cosx in our limit, we get [(1 - (1/2)x^2) - 1]/(x^2 * (1 - (1/2)x^2 + higher order terms)).
Step 6: Simplify the numerator. 1 - 1 = 0, so the numerator becomes -x^2/2.
Step 7: Simplify the denominator by multiplying out the terms and keeping only the highest degree of x. We get x^2 - (1/2)x^4 + higher order terms.
Step 8: Cancel out the common factor of x^2 in the numerator and denominator. Our expression now becomes -1/2 divided by (1 - (1/2)x^2 + higher order terms).
Step 9: Take the limit as x approaches 0. Since the higher order terms involve powers of x, as x approaches 0, those terms become negligible, leaving us with -1/2 divided by 1.
Step 10: Simplify the expression -1/2 divided by 1. This gives us -1/2.
Therefore, the limit of (secx-1)/(x^2) as x approaches 0 algebraically is -1/2.