a) Write each answer as an equality, and any decimals up to 7 decimal places. Clearly state the value of and δ (or M or N and δ) in each case. (i) How close to 4 do we need to take x so that (x/2 − 2) < 0.001 ? (ii) How close to 4 do we need to take x so that (x/2 − 2) > 0.0001 ? (iii) How close to 0 do we need to take x so that (2x + 9) < 9.0001 ? (iv) How close to 0 do we need to take x so that (2x + 9) > 8.999?
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
(a) Write each answer as an equality, and any decimals up to 7 decimal places. Clearly state the
value of and δ (or M or N and δ) in each case.
(i) How close to 4 do we need to take x so that (x/2
− 2) < 0.001 ?
(ii) How close to 4 do we need to take x so that (x/2
− 2) > 0.0001 ?
(iii) How close to 0 do we need to take x so that (2x + 9) < 9.0001 ?
(iv) How close to 0 do we need to take x so that (2x + 9) > 8.999?
(v) How close to 0 do we need to take x so that (x^2 + 6x + 9) < 9.001 ?
(vi) How close to 0 do we need to take x so that (x^2 + 6x + 9) < 9.0001 ?
(vii) How close to 0 do we need to take x so that (x^2 + 6x + 9) > 8.9999 ?
(viii) How close to −7 do we need to take x so that 1/(x+7)^4 > 10000 ?
(ix) How close to −7 do we need to take x so that 1/(x+7)^4 > 100000 ?
(x) How close to 0 do we need to take x so that ln x < −10000?
(b) Use the definition of a limit to show that
(i) limx→4
(x/2 − 2) = 0
(ii) limx→0
(2x + 9) = 9
(iii) limx→0
(x^2 + 6x + 9) = 9
(iv) limx→−7
1/(x+7)^4 = ∞
(v) limx→0+
lnx = −∞
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